"     ••;•**-:".   :': 

-  m  I   Hfl 

I^^H 


ELEMENTS 


OF 


WAVE    MOTION 


RELATING   TO 


SOUND    AND    LIGHT. 


ELEMENTS 


WAVE    MOTION 


RELATING   TO 


SOUND    AND    LIGHT. 


A    TEXT    BOOK 

PREPARED    EXPRESSLY    FOR    THE    USE    OF    THE 

CADETS    OF    THE    UNITED     STATES    MILITARY     ACADEMY, 

WEST     POINT. 


BY 

PETER     S.     MICHIE, 

LATE   PROFESSOR  OF   NATURAL    AND    EXPERIMENTAL   PHILOSOPHY    IN    THE    U.    S.    MILITARY 
ACADEMY  ;     AND    BREVET-LIEUTENANT-COLONEL   U.   S.    ARMY. 


THIRD 


NEW  YORK : 

JOHN    WILEY   &    SONS. 

LONDON:   CHAPMAN   &    HALL,    LIMITED. 

1904. 


Copyright,  1882,  by  D.  Van  Nostrand, 


ROBERT  DRUMMOND,    ELECTROTYPER   AND    PRINTER,    NEW   YORX. 


PREFACE. 


rriHIS  text-book,  as  is  stated  on  the  title-page,  has  been  prepared 
expressly  for  the  use  of  the  Cadets  of  the  United  States 
Military  Academy,  and  this  specific  object  has  therefore  wholly 
controlled  its  design  and  restricted  its  scope.  It  is  thus  in  no 
sense  a  treatise.  Because  of  the  limited  time  allotted  to  the  sub- 
jects of  sound  and  light  in  the  present  distribution  of  studies  at 
the  Academy,  the  problem  of  arranging  a  fundamental  course  of 
sufficient  strength,  to  be  something  more  than  popular,  and  yet  to 
be  mastered  within  the  allotted  time,  has  been  somewhat  perplex- 
ing. The  basis  of  this  arrangement  is  necessarily  the  mathematical 
attainments  of  the  class  for  which  the  course  is  intended.  In  this 
respect,  the  class  has  completed  the  study  of  elementary  Mathe- 
matics, as  far  as  to  include  the  Calculus,  and  has  had  a  four 
months'  study  of  the  application  of  pure  Mathematics,  in  a  course 
of  Analytical  Mechanics.  With  these  elements  to  govern,  this 
text-book  has  been  designed  for  a  seven  weeks'  course,  including 
advance  and  review.  The  fact  of  being  able,  through  the  discipline 
of  the  Academy,  to  exact  of  each  student  a  certain  number  of  hours 
of  hard  study  on  each  lesson,  is  of  course  an  important  element 
necessary  to  be  stated. 

The  study  of  the  text  is  supplemented  by  lectures,  in  which  the 

271630 


vi  PREFACE. 

principles  of  Acoustics  and  Optics  are  amply  illustrated  by  the  aid 
of  a  very  well  equipped  laboratory  of  physical  apparatus.  Carefully 
written  notes  of  the  lectures  are  submitted  by  each  student  to  the 
instructor  on  the  following  morning  for  revision  and  criticism. 
Important  errors  of  fact  and  misinterpretation  of  principle  are  thus 
at  once  detected,  corrected,  and  hence  prevented  from  obtaining  a 
lodgment  in  the  mind  of  the  pupil.  Another  element  in  this 
matter  of  instruction,  of  sufficient  importance  to  be  mentioned,  is 
the  opportunity  freely  exercised  by  each  student  of  making  known 
the  difficulties  that  he  has  encountered,  before  being  called  upon 
to  exhibit  his  proficiency  in  the  lesson  of  the  day.  It  is  required 
that  these  difficulties  shall  be  clearly  and  exactly  stated,  in  order 
that  the  instructor  may,  by  a  judicious  question  or  a  concise  expla- 
nation, enable  the  student  to  clear  up  the  difficulty  as  of  himself, 
and  thus  complete  the  elucidation. 

The  author  believes  that  this  method  of  instruction,  taken  as  a 
whole,  is  in  sufficiently  intimate  accord  with  the  text  as  to  gain  the 
following  advantages,  viz. :  1°,  the  tasks  are  of  the  requisite  strength 
to  demand  all  the  study-time  allotted  to  his  department  of  instruc- 
tion, and  thus  is  secured  the  invaluable  mental  effort  and  discipline 
due  to  a  specified  number  of  hours  of  hard  study ;  2°,  while  the 
daily  tasks  are  progressive,  they  are  based  on  fundamental  princi- 
ples which  require  the  exercise  of  a  rational  faith,  and  develop  a 
continual  growth  of  confidence  in  the  mind  of  the  pupil,  and  a 
belief  in  his  own  ability  to  overcome  each  difficulty  as  it  arises ; 
3°,  when  the  course  is  completed,  the  student  finds  himself 
equipped  with  a  satisfactory  knowledge  of  the  essential  principles 
of  the  physical  science,  to  which  he  may  add  by  further  individual 
study,  without  the  necessity  of  reconstructing  his  foundation. 


PREFACE.  vil 

The  elements  of  character  developed  in  the  student  by  this  sys- 
tem of  instruction,  viz.,  confidence  in  his  powers,  reliance  on 
individual  effort,  and  capacity  to  appreciate  truly  his  sources  of 
information,  are  of  essential  importance  in  a  career  where  he  may 
be  called  upon  in  emergencies  to  exercise  self-control,  and  to  meet 
manfully  unforeseen  difficulties ;  and  they  offer  a  sufficient  reason 
for  the  importance  given  to  these  studies  in  the  curriculum  of  the 
Academy. 

Text-books  are  generally  compilations.  The  subject-matter  of 
this  text  has  been  gathered  by  the  author  from  whatever  source 
appeared  to  him  best  for  the  purpose  in  view.  And  as  it  is  often 
desirable  to  refer  to  original  treatises,  for  a  better  conception  of  the 
subject  under  discussion,  a  list  of  authors  is  appended  to  this 
Preface. 

In  the  arrangement  of  the  matter,  the  author  has  been  governed 
alone  by  the  necessities  of  the  case  and  the  restrictions  of  the 
course.  It  has  therefore  seemed  advisable  to  arrive  at  the  deduc- 
tion of  Fresnel's  wave  surface  as  expeditiously  as  possible,  and  on 
the  way  to  establish  all  of  the  essential  principles  of  undulatory 
motion  common  to  sound  and  light.  Sufficient  theoretical  atten- 
tion is  paid  in  the  text  to  the  wave  surface,  and  a  study  of  its  model 
in  the  lecture-room  makes  clear  its  important  properties  and  those 
of  its  special  cases.  Acoustics  is  briefly  treated,  and  is  indeed  made 
subsidiary  to  Optics,  by  utilizing  its  numerous  illustrations  in  vi- 
bratory motion,  so  that  the  laws  of  this  motion  may  be  the  more 
clearly  apprehended  in  the  subject  of  light.  In  Optics,  while  the 
essential  principles  of  the  deviation  of  light  by  lenses  and  mirrors, 
the  construction  of  optical  images  and  the  principal  telescopic  com- 
binations, are  carried  only  to  first  approximations,  and  are  some- 


viii  PREFACE. 

what  more  condensed  than  is  usual,  nothing  essential  to  the 
Academic  course  of  Astronomy  has  been  omitted.  The  part 
relating  to  physical  Optics  is  very  concise,  but  the  experiments 
performed  and  illustrations  given  in  the  lecture  room,  especially  in 
diffraction,  dispersion,  and  polarization,  largely  remedy  this  defect. 

The  figures  throughout  the  text  were  drawn  by  Lieut.  Arthur 
Murray,  1st  U.  S.  Artillery,  Acting  Asst.  Professor  of  Philosophy, 
U.  S.  M.  A.,  to  whom  I  desire  to  acknowledge  my  great  indebted- 
ness. 

P.  S.  M. 

WEST  POINT,  N.  Y.,  May,  1882. 


LIST    OF    AUTHORITIES. 


1.  AIKY.    UNDULATORY  THEORY  OF  OPTICS.    1866. 

2.  ANNALES  DE  CHIMIE  ET  DE  PHYSIQUE.    Tome  XXXVI. 

3.  ARCHIVES  DE  MUSEE  TEYLER.    Vol.  I.     1868. 

4.  BARTLETT.    ANALYTICAL  MECHANICS.    1858. 

5.  BILLET.    TRAITE  D'OPTIQUE  PHYSIQUE.    1859. 

6.  CHALLIS.    LECTURES  ON  PRACTICAL  ASTRONOMY.    1879. 

7.  CODDINGTON.    SYSTEM  OF  OPTICS.     1829. 

8.  COMPTES  RENDUS.    Vol.  LXVI.     1863. 

9.  DAGUIN.     TRAITF,  DE  PHYSIQUE.    Vols.  I  and  IV.    1879. 

10.  DONKIN'S  ACOUSTICS.    1870. 

11.  ENCYCLOPEDIA  BRITANNICA.    Ninth  Edition,  Vol.  VII.    1877. 

12.  EVERETT'S  UNITS  AND  PHYSICAL  CONSTANTS.    1879. 

13.  FRESNEL.    CEUVRES  DE.    Vols.  I,  II. 

14.  HELMHOLTZ.    TONEMPFINDUNGEN.    1866. 

15.  HELMHOLTZ.    PHYSIOLOGISCHE  OPTIK.    1867. 

16.  JAMIN.    COURS  DE  PHYSIQUE.     Vol.  III.     1866. 

17.  LAME.    THEORIE  MATHEMATIQUE  DE  L'ELASTICITE.    1852. 

18.  LLOYD.    WAVE  THEORY  OF  LIGHT.    1873. 

19.  PARKINSON'S  OPTICS.    Third  Edition.     1870. 

20.  POTTER'S  OPTICS.    1851. 

21.  PRICE.    INFINITESIMAL  CALCULUS.    Vol.  IV.    1865. 

22.  ROOD.    MODERN  CHROMATICS.     1879. 

23.  THOMSON  AND  TAIT.    NATURAL  PHILOSOPHY.    Vol.  I.    1879. 

24.  VERDET.    COURS  DE  PHYSIQUE.    Vol.  II.    1869. 

25.  VERDET.    LEQONS  DE  L'OPTIQUE  PHYSIQUE.    Vols.  I  and  II. 


CONTENTS. 


PART    I.     WAVE    MOTION. 

AST.  PAGH 

1.  General  Equation  of  Energy , 17 

2.  Relation  of  Hypothesis  to  Theory 17 

3-9.  Molecular  Science 18 

ELASTICITY. 

10-15.  Elasticity,  General  Definitions 19 

16-20.  Origin  of  Theory  of  Elasticity 20 

21-22.  Elastic  Force  denned 22 

23-24.  Elasticity  of  Solids 23 

25-27.  Fundamental  Coefficients  of  Elasticity 25 

28-37.  Analytical  Expression  of  Elastic  Forces  developed  in  the  Motion 

of  a  System  of  Molecules  subjected  to  Small  Displacements.. .  27 

38-43.  Surfaces  of  Elasticity 35 

WAVES. 

44-51.  General  Definitions  and  Form  of  Function 38 

52.  Simple  Harmonic  Motion 40 

53-55.  The  Harmonic  Curve 41 

56-61.  Composition  of  Harmonic  Curves 42 

62-64.  Wave  Function 45 

65-66.  Wave  Interference 46 

67-68.  Interference  of  any  Number  of  Undulations 48 

69-71.  The  Principle  of  Huyghens 49 

72.  Diffusion  and  Decay  of  Kinetic  Energy 52 

73-74.  Reflection  and  Refraction 53 

75-76.  Diverging,  Converging,  and  Plane  Waves 53 


CONTENTS.  xi 

ABT.  PAGB 

77-80.  Reflection  and  Refraction  of  Plane  Waves 54 

61-87.  General  Construction  of  the  Reflected  and  Refracted  Waves 56 

88-92.  Utility  of  Considering  the  Propagation  of  the  Disturbance  by 

Plane  Waves 59 

93-96.  Relation  between  the  Velocity  of  Wave  Propagation  of  Plane 

Waves  and  the  Wave  Length  in  Isotropic  Media 63 

97-102.  Plane  Waves  in  a  Homogeneous  Medium  of  Three  Unequal  Elas- 
ticities in  Rectangular  Directions 66 

103-105.  The  Double-Napped  Surface  of  Elasticity 69 

106-110.  The  Wave  Surface 71 

111-112.  Construction  of  the  Wave  Surface  by  Means  of  Ellipsoid  (W) 75 

113-115.  Relations  between  the  Directions  of  Normal  Propagation  of  Plane 
Waves,  the  Directions  of  Radii -Vectores  of  the  Wave  Surface, 

and  the  Directions  of  Vibrations 77 

116-125.  Discussion  of  the  Wave  Surface 79 

126-128.  Relations  between  the  Velocities  and  Positions  of  Plane  Waves 

with  respect  to  the  Optic  Axes 84 

129-134.  Relations  between  the  Velocities  of  Two  Rays  which  are  Coinci- 
dent in  Direction  and  the  Angles  that  this  Direction  makes 
with  the  Axes  of  Exterior  Conical  Refraction. . .  86 


PART    II.      ACOUSTICS. 

135-140.  Acoustics  defined  ;  Description  of  Ear  ;  Curve  of  Pressure 89 

141-142.  Propagation  of  a  Disturbance  in  an  Indefinite  Cylinder 91 

143-149.  Curve  of  Pressure  due  to  a  Tuning-Fork 93 

150.  Illustrations  of  Motion  transmitted 96 

151-154.  Properties  of  Sound ;  Intensity,  Pitch,  Quality 95 

155-158.  Curves  of  Pressure  for  Noises 97 

159-161.  Curves  of  Pressure  for  Simple  Tones 99 

162-165.  Properties  of  Audition  ;  Definitions  of  Simple,  Single,  and  Musi- 
cal Tones 101 

166-167.  Musical  Intervals 103 

16&-169.  Musical  Scales 104 

170.  Perfect  Accords. . .                                              106 


Xll  CONTENTS. 

ART.  PAGE 

171-174.  The  Diatonic  Scale  ;  Harmonics 106 

175-178.  Sympathetic  Resonance 109 

179-182.  Separation  of  Vibrating  Bodies  into  Component  Parts Ill 

183-184.  Velocity  of  Sound  in  any  Isotropic  Medium 112 

185-189.  Velocity  of  Sound  in  Gases '. . . .   11$ 

190.  Pressure  of  a  Standard  Atmosphere 116 

191.  Height  of  the  Homogeneous  Atmosphere 116 

192-194.  Final  Velocity  of  Sound  in  Air 117 

195.  Formula  for  Velocity  of  Sound  in  any  Gas 118 

196-199.  Velocity  of  Sound  in  Air  and  other  Gases  as  Affected  by  their  not 

being  Perfect  Gases 119 

200.  Velocity  of  Sound  in  Gases  independent  of  the  Barometric  Pres- 
sure    121 

201-203.  Velocity  of  Sound  in  Liquids 122 

204r-209.  Velocity  of  Sound  in  Solids 124 

210.  Reflection  and  Refraction  of  Sound 126 

211.  Consequences  of  the  Laws  of  Reflection 127 

212.  Refraction  of  Sound 128 

213-221.  General  Equations  for  the  Vibratory  Motion  of  a  Stretched  String  128 
222-225.  Application  of  Equations  to  the  Longitudinal  Vibration  of  a  Rod.  136 

226-228.  Vibrations  of  Air  Columns ;  1°,  Closed  at  One  End 138 

229-230.  2°.  Open  Air  Columns 140 

231-232.  Relative  Velocities  of  Sound  in  Different  Material 142 

233-234.  Transversal  Vibration  of  Elastic  Rods 143 

235.  Harmonic  Vibrations  of  Elastic  Rods 145 

236.  Tuning  Forks 146 

237.  Vibration  of  Plates 147 

238.  Vibration  of  Membranes 14& 

239-240.  Subjects  referred  to  by  Lecture 149 


PART    III.      OPTICS. 

241-242.  Light  and  Optics  defined  ;  Geometrical  and  Physical  Optics ISC' 

243-245.  Luminif erous  Ether  and  its  accepted  Properties 150 

246-247.  Definitions 151 

248-249.  Properties  of  Bodies  in  Regular  Reflection 152- 


CONTENTS.  xiil 

ABT.  PAGE 

249-250.  Medium  defined  ;  Opaque  and  Transparent 153 

251-252.  Shadows  and  Shade 153 

253-256.  Photometry 154 

257.  Velocity  of  Light  by  Eclipse  of  Jupiter's  Satellite 157 

258.  Velocity  of  Light  by  Aberration 158 

259-261.  Velocity  of  Light  by  Actual  Measurement 158 


GEOMETRICAL    OPTICS. 

262-263.  Index  of  Refraction  ;  Radiant  and  Focus 160 

264-265.  Deviation  of  Light  by  Plane  Surfaces 161 

266-267.  Refraction  by  Optical  Prisms 163 

268-270.  Deviation  of  Light  by  Spherical  Surfaces  165 

271.  Application  of  Eq.  (305)  to  Reflection  at  Plane  Surfaces 168 

272.  Multiple  Reflection— Parallel  Mirrors 169 

273.  Multiple  Reflection  by  Inclined  Mirrors 169 

274-275.  Angular  Velocity  of  the  Reflected  Ray 170 

276-277.  Deviation  of  Small  Direct  Pencils  by  Spherical  Surfaces 170 

278.  Lenses 172 

279.  Principal  Focal  Distance  of  a  Lens 173 

280.  Discussion  of  the  Properties  of  a  Lens 173 

281.  Relative  Velocities  of  the  Radiant  and  Focus 175 

282-284.  Discussion  of  Spherical  Reflectors 175 

285.  Power  of  a  Lens 177 

286-287.  To  find  the  Principal  Focal  Distance  of  a  Lens  and  a  Reflector.. . .  178 

288.  Deviation  of  Oblique  Pencils  by  Reflection  or  Refraction  at  Spher- 

ical Surfaces 180 

289.  Circle  of  Least  Confusion 182 

290-291.  The  Positions  of  the  Foci 182 

292.  Particular  Cases  of  Caustic  Curves 184 

293.  Critical  Angle  for  Refraction 186 

294-295.  Spherical  Aberration 188 

296.  Optical  Centre 191 

297.  Focal  Centres  of  a  Lens 192 

298.  The  Eye 193 

299.  Vision 195 

300-303.  Optical  Images 196 


Xiv  CONTENTS. 

ART.  PAGE 

304.  Optical  linages  discussed  for  a  Concave  Lens 199 

305-306.  Optical  Images  discussed  for  a  Convex  Lens 200 

307.  Optical  Instruments 201 

308.  Camera  Lucida „ 201 

309.  Camera  Obscura 202 

310.  Solar  Microscope 202 

811.  Microscopes 208 

312.  Magnifying  Power  of  a  Microscope 204 

313.  The  Field  of  View  of  a  Microscope 205 

314.  Oculars 206 

315.  Compound  Microscope 208 

316.  Telescopes 209 

317-319.  The  Astronomical  Refracting  Telescope 209 

320.  The  Galilean  Telescope 211 

321.  The  Terrestrial  Telescope 212 

322.  Reflecting  Telescopes 212 

323.  The  Newtonian  Telescope 213 

324.  The  Gregorian  Telescope 213 

325.  The  Cassegrainian  Telescope 214 

326-328.  The  Magnifying  Power  of  any  Combination  of  Lenses  and  Mir- 
rors   215 

329.  Brightness  of  Images 218 

330.  Magnifying  Power  of  the  Principal  Telescopes 219 


PHYSICAL     OPTICS. 

331-333.  Solar  Spectrum 219 

334.  Dispersion 221 

335.  Frauenhofer's  Lines 221 

336.  Normal  Spectrum 222 

337-338.  Absence  of  Dispersion  in  Ether  of  Space 222 

339.  Irrationality  of  Dispersion 223 

340-341.  Color 224 

342-345.  Absorption 225 

346.  Emission 227 

347.  Spectrum  Analysis 228 

348.  Spectroscopes  and  Spectrometers 228 


CONTENTS.  XV 

ABT.  PAGE 

349-350.  Absorption  Spectra 229 

351.  Color  and  Intensity  of  Transmitted  Light 229 

352-353.  Color  of  Bodies  ;  Constants  of  Color ;  Mixed  Colors 230 

354.  Phosphorescence  and  Fluorescence 231 

355-356.  Achromatism 231 

357.  Achromatism  of  Prisms 232 

358.  Table  of  Refractive  Indices 234 

359.  Achromatism  of  Lenses 234 

360-362.  Chromatic  Aberration  of  a  Lens 235 

363-369.  Rainbow— Cartesian  Theory 236 

370.  Rainbow— Airy's  Theory 240 

371-378.  Interference  of  Light 242 

379-381.  Colors  of  Thin  Plates 248 

382-388.  Diffraction 250 

389.  Table  of  Wave  Lengths  of  Fixed  Lines  in  the  Normal  Spectrum.  255 

390-392.  Polarized  Light 256 

393-396.  Polarization  by  Double  Refraction 257 

397.  Polarization  by  Reflection 260 

398.  Polarization  by  Refraction .260 

399-400.  Contrasts  between  Natural  and  Polarized  Light 261 

401-402.  Analytical  Proof  of  the  Trans  versality  of  the  Molecular  Vibra- 
tions    262 

403-404.  Distinction  between  Natural  and  Polarized  Light 265 

405.  Construction  of  Refracted  Ray  in  Isotropic  Media 266 

406.  Uniaxal  Crystals 267 

407-409.  Construction  of  Refracted  Ray  in  Uniaxal  Crystals 268 

410.  Biaxal  Crystals 269 

411.  Interior  Conical  Refraction 269 

412.  Exterior  Conical  Refraction 270 

413.  Mechanical  Theory  of  Reflection  and  Refraction 270 

414.  Reflection  of  Light  polarized  in  Plane  of  Incidence 271 

415.  Reflection  of  Light  polarized  Perpendicular  to  Plane  of  Incidence  272 

416.  Intensity  of  Reflected  and  Refracted  Light 273 

417.  Reflection  of  Natural  Light .  Brewster's  Law 273 

418.  Change  of  Plane  of  Polarization  by  Reflection 275 

419.  Elliptic  Polarization 276 

420-421.  Reflection  of  Polarized  Ray  at  Surface  separating  Denser  from 

Rarer  Medium .  277 


xvi  CONTENTS, 

ART.  PAGE 

432.  Fresnel's  Rhomb 279 

423-424.  Plane,  Elliptical,  and  Circularly  Polarized  Light  by  Fresnel's 

Rhomb 280 

425-427.  Interference  of  Polarized  Light 282 

428-429.  Colored  Rings  in  Uniaxal  Crystals 284 

430.  Colored  Curves  in  Biaxal  Crystals 286 

431.  Accidental  Double  Refraction 287 

432.  Rotatory  Polarization 287 

433.  Transition  Tint 287 

434.  Cause  of  Rotatory  Polarization 288 

435.  Saccharimetry 288 

Conclusion .  288 


PART    I. 
WAVE      MOTION. 


1.  Equation  (E)  of  Analytical  Mechanics  (Michie), 


expresses  in  mathematical  language  the  law  that  the  potential 
energy  expended  is  equal  to  the  kinetic  energy  developed.  Every 
analytical  discussion  of  the  action  of  force  upon  matter  must 
be  founded  upon  this  general  equation.  For  the  complete  solution 
of  every  problem  of  energy,  it  is  necessary  to  know  the  intensities, 
lines  of  action,  and  points  of  application  of  the  acting  forces,  the 
masses  acted  Upon,  and  to  possess  a  perfect  mastery  of  such  mathe- 
matical processes  as  are  necessary  to  pass  to  the  final  equations 
whose  interpretation  will  make  known  the  effects.  These  difficul- 
ties, which,  in  Mechanics,  limit  the  discussion  to  the  free,  rigid 
solid,  and  to  the  perfect  fluid,  are,  in  Molecular  Mechanics,  almost 
insuperable;  since  we  neither  know  the  nature  of  the  forces  which 
unite  the  elements  of  a  body  into  a  system,  nor  the  constitution  of 
the  elements  themselves. 

2.  But  the  faculty  of  observation,  being  cultivated  and  logically 
directed,  has  enabled  scientific  men  to  originate  experiments  which, 
because  of  our  inherent  faith  in  the  uniformity  of  the  laws  of  na- 
ture, have  resulted  in  certain  hypotheses  as  to  the  nature  of  sound, 
light,  heat,  and  other  molecular  sciences.  When  an  hypothesis  not 
only  satisfactorily  explains  the  known  phenomena  of  the  science  in 
question,  but  even  predicts  others,  it  then  becomes  a  theory,  and 
its  acceptance  is  more  or  less  complete.  An  hypothesis  is  related 
to  a  theory  as  the  scaffolding  to  the  structure,  the  latter  being  so 
proportioned  in  all  its  parts  as  to  be  in  the  completest  harmony, 
while  the  former  may  be  modified  in  any  way  to  suit  the  ever- 
varying  necessities  of  the  architect. 


18  ELEMENTS    OF    WAVE    MOTION. 

While  there  are  many  matters  concerning  which  a  reasonable 
doubt  may  be  entertained,  because  of  insufficient  data,  the  progress 
of  scientific  thought  and  the  fertility  of  scientific  research  have, 
within  recent  times,  estabi  shed  certain  facts  that  are  now  univer- 
sally accepted. 

3.  Molecular  Science*     Molecular  science  is  a  branch  of 
Mechanics  in  which  the  forces  considered  are  the  attractions  and 
repulsions  existing  among  the  molecules  of  a  body,  and  the  masses 
acted  upon  are  the  indefinitely  small  elements,  called  molecules,  of 
which  the  body  is  composed.     It  embraces  light,  heat,  sound,  elec- 
trics, and,  in  one  sense,  chemistry. 

4.  From  the  facts  of  observation  and  experiment,  it  is  assumed 
that  all  matter,  whether  solid,  liquid,  or  gaseous,  is  made  up  of  an 
innumerable  number  of  molecules  in  sensible,  though  not  in  actual 
contact ;  that  these  molecules  are  so  small  as  not  to  be  within  range 
of  even  our  assisted  vision  ;  and  that  they  are  separated  from  each 
other  by  distances  which  are  very  great  compared  with  their  actual 
linear  dimensions. 

5.  The  molecular  forces,  which  determine  the  particular  state 
of  the  matter,  are  either  attractive  or  repulsive.     When  the  attrac- 
tive forces  exceed  the  repulsive  in  intensity,  the  body  is  a  solid; 
when  equal  to  the  repulsive,  a  liquid ;  and  when  less,  a  gas.     The 
relative  places  of  equilibrium  of  the  molecules  are  determined  by 
the  molecular  forces  called  into  play  by  the  action  of  extraneous 
forces  applied  to  the  body.     Thus,  when  a  solid  bar  is  subjected  to 
the  action  of  an  extraneous  force,  either  to  elongate  or  to  compress 
it,  the  molecules  assume  new  positions  of  equilibrium  with  each 
increment  of  force,  and,  in  either  case,  the  aggregate  molecular 
forces  developed  are  equal  in  intensity,  but  contrary  in  direction, 
to  the  extraneous  force  applied.     In  general,  where  rupture  does 
not  ensue,  the  extraneous  forces  applied  are  much  less  than  the 
molecular  forces  capable  of  being  called  into  play. 

6.  While  we  are  ignorant  of  the  true  nature  of  force  and  mat- 
ter, our  senses  enable  us  to  appreciate  the  effects  of  the  former 
upon  the  latter.     Our  whole  knowledge  of  the  physical  sciences  i& 
based  upon  the  correct  interpretation  of  these  sensuous  impressions. 
Observation  teaches  that  if  a  body  be  subjected  to  the  action  of  an 
extraneous  force,  the  effect  of  the  force  is  transmitted  throughout 


RELATING    TO    SOUND    AND    LIGHT.  19 

the  body  in  all  directions,  and  since  the  body  is  connected  with  the 
rest  of  the  material  universe,  there  is  no  theoretical  limit  to  the 
ultimate  transfer  of  this  effect  throughout  space. 

7.  Among  the  appreciable  effects  of  force  are  the  changes  of 
state  with  respect  to  rest  and  motion.  These  can  be  transferred 
from  an  origin  to  another  point  in  but  two  ways,  viz. : 

1°.  By  the  simultaneous  transfer  of  the  body,  which  is  the  de- 
pository of  the  motion. 

2°.  By  the  successive  actions  and  reactions  between  the  consec- 
utive molecules  along  any  line  from  the  origin. 

In  the  molecular  sciences,  the  latter  is  assumed  to  be  the  method 
of  transfer,  and  the  object  of  the  succeeding  discussion  is  to  inves- 
tigate the  nature  of  the  disturbance,  the  circumstances  of  its 
progress,  and  the  behavior  of  the  molecules  as  they  become  involved 
in  it. 

8.  While   the   initial   disturbance    is    perfectly   arbitrary,    the 
molecular  motions  produced  through  its  influence  in  any  medium 
are  necessarily  subjected  to  the  variable  conditions  which  result 
from  the  action  of  the  forces  that  unite  the  molecules  into  a  mate- 
rial system.     The  problems  are  then  those  of  constrained  motion. 

9.  Among  the  physical  properties  of  bodies,  elasticity  is  of  such 
great  importance,  that  a  complete  knowledge  of  its  mathematical 
theory  is  essential  to  the  thorough  elucidation  of  many  of  the  phe- 
nomena of  molecular  science.     The  limits  of  this  text  permit  but  a 
passing  allusion  to  its  more  important  laws. 

ELASTICITY. 

10.  A  body  is  said  to  be  homogeneous  when  it  is  formed  of 
similar  molecules,  either  simple   or  compound,    occupying  equal 
spaces,  and  having  the  same  physical  properties  and  chemical  com- 
position.    In  such  a  body,  a  right  line  of  given  length  I  and  deter- 
minate direction  is  understood  to  pass  through  the  same  number  n 

of  molecules  wherever  it  is  placed  ;  the  ratio  -  will  vary  with  the 

W/ 

direction  of  I.  In  crystalline  bodies,  considered  as  homogeneous, 
-  varies  with  the  direction  ;  in  homogeneous  non-crystalline  bodies, 
such  as  glass,  the  ratio  varies  insensibly,  or  is  independent  of  the 


20  ELEMENTS   OF    WAVE    MOTION. 

direction.     This  supposition  requires  n  to  be  very  great,  however 
small  I  may  be. 

11.  That  property,  by  which  the  internal  forces  of  a  body  or 
medium  restore,  or  tend  to  restore,  the  molecules  to  their  primitive 
positions,  when  they  have  been  moved  from  these  positions  by  the 
action  of  some  external  force,  is  called  Elasticity. 

12.  The  elasticity  is  said  to  be  perfect  when  the  body  always 
requires  the  same  force  to  keep  it  at  rest  in  the  same  bulk,  shape, 
and  temperature,  through  whatever  variations  of  bulk,  shape,  and 
temperature  it  may  have  been  subjected. 

13.  Every  body  has  some  degree  of  elasticity  of  bulk.     If  a  body 
possess  any  degree  of  elasticity  of  shape,  it  is  called  a  solid ;  if  none, 
a  fluid.     All   fluids  possess  great  elasticity  of  bulk.     While  the 
elasticity  of  shape  is  very  great  for  many  solids,  it  is  not  perfect  for 
any.     The  degree  of  distortion  within  which  elasticity  of  shape  is 
found,  is  essentially  limited  in  every  solid  ;  when  the  distortion  is 
too  great,  the  body  either  breaks  or  receives  a  permanent  set ;  that 
is,  such  a  molecular  displacement  that  it  does  not  return  to  its 
original  figure  when  the  distorting  force  is  removed. 

14.  The  limits  of  elasticity  of  metal,  stone,  crystal,  and  wood 
are  so  narrow  that  the  distance  between  any  two  neighboring  mole- 
cules of  the  substance  never  alters  by  more  than  a  small  proportion 
of  its  own  amount,  without  the  substance  either  breaking  or  expe- 
riencing a  permanent  set.     In  liquids,  there  are  no  limits  of  elas- 
ticity as  regards  the  magnitude  of  the  positive  pressures  applied ; 
and  in  gases,  the  limits  of  elasticity  are  enormously  wider  with 
respect  to  rarefaction  than  in  either  solids  or  liquids,  while  there  is 
a  definite  limit  in  condensation  when  the  gas  is  near  the  critical 
temperature. 

15.  The  substance  of  a  homogeneous  solid  is  called  isotropic 
when  a  spherical  portion  exhibits  no  difference,  in  any  direction,  in 
quality,  when  tested  by  any  physical  agency.     When  any  difference 
is  thus  manifested,  it  is  said  to  be  celotropic. 

16.  Origin  of  the  Theory  of  Elasticity.    In  Mechan- 
ics, by  supposing  the  bodies  perfectly  rigid,  and  the  distances  of 
the  points  of  application  of  the  extraneous  forces  invariable,  how- 
ever great  the  forces,  the -problems  are  much  simplified,  without 
affecting  their  generality.     But  this  ignores  the  law  by  which  the 


RELATING    TO    SOUND    AND    LIGHT.  21 

reciprocal  influence  is  transmitted  from  point  to  point  of  the  body, 
and  by  which  the  action  of  one  force  is  counterbalanced  by  the 
actions  of  others.  In  reality,  the  body  undergoes  deformation,  and 
when  the  limit  is  reached,  rupture  ensues.  The  mathematical 
theory  has  arisen  from  the  necessity  of  a  knowledge  whereby  these 
permanent  deformations  and  rupture  may  be  avoided.  This  theory 
has  been  extended  to  the  determination  of  the  laws  of  small  motions, 
or,  in  general,  to  the  vibrations  of  elastic  media. 

17.  The  initial  state  of  a  homogeneous  body  is  considered  to  be 
that  in  which  it  is  perfectly  free  from  all  extraneous  forces,  to  be, 
indeed,  that  of  a  body  falling  freely  in  vacuo.     Such  a  body  is  then 
the  geometrical  place  of  an  innumerable  number  of  material  points, 
which  are  distinguished  from  the  rest  of  space  by  several  mechani- 
cal properties.     Each  of  these  material  points  is  called  a  molecule. 

18.  When  such  a  body  is  subjected  to  the  action  of  an  extra- 
neous force,  either  a  tension  or  a  pressure,  a  motion  of  its  surface 
particles  ensues,  and  this  disturbance  is  propagated  to  the  interior 
molecules  ;  the  body  becomes  slightly  distorted,  and  soon  takes  a 
new  state  of  equilibrium.     When  the  external  forces  are  removed, 
the  internal  forces  are  again  balanced,  and  the  original  condition  is 
restored,  provided  there  is  no  permanent  set.     All  changes  of  form 
of  a  solid,  or  any  variation  of  the  relative  distances  of  its  material 
points,  are  ever  accompanied  by  the  development  of  attractive  or 
repulsive  forces  between  the  molecules.     These  variations  and  forces 
begin,  increase,  decrease,  and  end  at  the  same  time,  and  hence  are 
mutually  dependent. 

19.  The  properties  of  a  solid  body  depending  only  upon  those 
of  its  material  points,  they  alone  are  the  foci  whence  emanate  these 
interior  forces. 

20.  Let  an  extraneous  force  be  applied  to  a  body,  and  consider 
its  effect  upon  any  two  molecules  sufficiently  near  each  other  to  be 
mutually  affected  by  their  changes  of  position.     Should  one  of  the 
molecules,  on  account  of  this  exterior  action,  approach  the  other,  a, 
mutual  repulsion  takes  place,  which,  in  time,  overcomes  the  motion 
of  the  first  molecule,  and  causes  the  second  to  take  its  new  position 
of  equilibrium  with  respect  to  the  first.     The  reverse  is  the  case 
when  the  first  molecule  withdraws  from  the  second,  and  an  attrac- 
tive force  is  developed  between  them.     If  r  represent  the  primitive 
distance,  Ar  may  represent  the  displacement.     Then  the  intensity 


22  ELEMENTS    OF    WAVE    MOTION. 

of  the  attractive  or  repulsive  force  developed  between  the  molecules 
may  be  represented  by  /(r,  Ar).  This  function  becomes  zero  when 
Ar  is  zero,  whatever  r  may  be ;  it  decreases  rapidly  when  r  has  a  sensi- 
ble value,  whatever  Ar  may  be,  since  all  cohesion  ceases  between 
two  parts  of  the  same  body  separated  by  an  appreciable  distance. 
Assuming  that  the  intensity  of  the  molecular  forces  varies  directly 
with  the  degree  of  displacement,  this  limitation  embodies  only  the 
cases  where  the  changes  of  form  are  very  small,  whether  the  extra- 
neous forces  are  extremely  small  or  the  bodies  considered  have 
great  rigidity.  Hence,  f  (r,  Ar)  is  limited  to  the  product  of  a 
function  of  r  and  the  first  power  of  Ar,  which  becomes  infinitely 
small  when  Ar  becomes  infinitely  small. 

21.   Elastic  Force  defined.    From  any  molecule  M  in  the 
interior  of  a  solid,  with  a  radius  equal  to  the  greatest  distance  be- 
yond which  /(/•)  is  insensible,   describe 
a  sphere.     This  volume  will  embrace  all 
molecules  that  influence  the  molecule  M, 
and  may  be  called  the  sphere  of  molecular 
activity.     Pass   a   plane   through   M,  di- 
viding the  sphere  into  the  two  parts  SAC 
and  SBC.     Normal  to  KN"  and  having  for 
its   base    a  differential    surface    <•>,    con- 
ceive a  cylinder  in  the  hemisphere  SBC.  p,  ure  { 
When  the  equilibrium  is  disturbed,  the 

molecules  in  SAC  will  act  on  the  molecules  of  the  cylinder. 
The  resultant  uE  of  all  these  actions  is  called  the  elastic  force 
exerted  by  SAC  upon  SBC,  referred  to  the  infinitesimal  sur- 
face w.  Integrating  this  function  with  respect  to  the  plane,  we 
obtain  the  elastic  force  referred  to  the  circle  SMC.  The  resultant 
uE  will,  in  general,  be  oblique  to  the  plane  element  w.  If  it  is 
normal  to  this  element  and  directed  towards  the  hemisphere  SAC, 
it  will  be  a  traction ;  if  normal  and  directed  toward  SBC,  it  will 
be  a  pressure;  if  parallel  to  the  plane  SMC,  it  will  be  the  tan- 
gential elastic  force. 

Similarly,  if  the  cylinder  is  situated  in  the  hemisphere  SAC,  the 
resultant  elastic  force  exerted  upon  the  molecules  of  the  cylinder 
by  the  molecules  in  SBC  is  represented  by  uE',  referred  to  the 
same  elementary  surface  w.  If  the  body,  slightly  changed  in  form, 


RELATING    TO    SOUND    AND    LIGHT.  23 


is  in  equilibrium  of  elasticity,  the  two  elastic  forces  uE  and 
should  be  equal  in  intensity,  but  contrary  in  direction.  Both, 
however,  will  represent  either  pulls,  pressures,  or  tangential  forces  ; 
that  is,  if  one  is  a  pull,  the  other  will  be  a  pull  directly  opposed 
to  it. 

The  elastic  force  uE,  considered  with  reference  to  the  element 
planes  w  drawn  parallel  to  each  other  through  all  points  of  the 
body,  will  vary  in  intensity  and  direction  from  point  to  point  ;  and 
at  the  same  point  M  will  vary  with  the  orientation  of  the  element 
plane  w. 

22.  The  direction  of  the  planes  w  may  be  determined  by  that  of 
their  normals.     Using  the  angles  0  and  V  to  designate  the  latitude 
and  longitude  of  the  point  where  the  normal  pierces  the  surface  of 
the  sphere  of  activity,  and  representing  by  x,  y,  and  z  the  co-ordi- 
nates of  this  point  referred  to  the  co-ordinate  axes,  we  have 

x  =.  cos  (^  cos  V>,        y  =  cos  0  sin  i/>,        z  =  sin  0. 

Representing  the  orthographic  projections  of  uE  by  wJT",  wl7", 
and  uZ  upon  the  co-ordinate  axes,  we  see,  in  the  case  of  equilibrium 
of  elasticity,  that  <*>E  will  be  a  function  of  the  five  variables  x,  y,  z, 
0,  and  i/>  ;  and  if  the  motion  be  progressive,  the  variable  t  will  also 
enter.  X,  Y,  and  Z  can  be  determined  from  uE,  (f>,  and  V>  ;  and, 
reciprocally,  the  latter  from  the  former.  X,  Y,  and  Z  are,  how- 
ever, usually  determined,  and  are,  in  general,  functions  of  the  six 
variables  (x,  y,  2,  0,  i/>,  £),  and  which  being  found  according  to  the 
special  circumstances  that  cause  the  deformation  of  the  body,  would 
enable  us  to  ascertain,  at  each  instant  and  at  each  point  of  the 
body,  the  direction  and  intensity  of  the  elastic  force  exerted  upon 
every  element  plane  passing  through  the  given  point.  In  brief,  the 
determination  of  these  functions  and  the  study  of  their  properties 
are  the  principal  objects  of  the  mathematical  theory  of  elasticity. 

23.  Elasticity  of  Solids.    Experiment  has  shown  that, 
when  a  solid  bar  is  subjected  to  small  elongations,  or  those  within 
elastic  limits,  the  following  laws  are  verified,  viz.  :  1°,  the  elonga- 
tions are  directly  proportional  to  the  length  of  the  bar;  2°,  they  are 
inversely  proportional  to  the  area  of  cross  section  ;  3°,  they  are 
directly  proportional  to  the  intensity  of  the  elongating  force  ;  4°, 


24 


ELEMENTS    OF    WAVE    MOTION. 


they  are  variable  for  bars  of  different  materials, 
mental  laws  can  be  expressed  by  the  equation, 

1     PI 


These  experi- 


(i) 


in  which  I  is  the  length  of  the  bar  unloaded,  s  the  area  of  cross- 
section,  P  the  intensity  of  the  stretching  force,  M  a  coefficient 
varying  with  the  nature  of  the  material,  and  A  is  the  correspond- 
ing elongation.  Making  s  =  1,  A  —  I,  we  get,  from  the  above 
equation,  P  =  M .  If,  therefore,  the  law  of  the  elongation 
should  remain  true  for  all  intensities,  M  would  be  that  intensity 
which,  applied  to  a  bar  of  unit  area  in  cross-section,  would  make 
the  elongation  equal  to  the  original  length.  Such  an  hypothesis 
gives  us  the  value  of  the  coefficient  M,  which  can  be  used  within 
the  limits  of  experiment.  M  is  called  the  coefficient  or  modulus  of 
longitudinal  elasticity,  or  Young's  modulus.  While  we  cannot 
experiment  over  such  wide  limits  in  longitudinal  compression, 
because  of  the  liability  to  flexure,  the  same  laws  are  held  to  be 
applicable,  with  the  same  limitations.  Taking  the  metre  for  the 
unit  of  length,  the  square  centimetre  for  the  unit  of  area,  and  the 
gramme  for  the  unit  of  intensity,  the  moduli  of  longitudinal  elas- 
ticity for  the  principal  metals  are,  according  to  Wertheim,  as  follows: 


Lead, .  177  xlO6 

Gold,  ......  813  xlO6 

Silver, 736  xlO6 

Zinc,   .    .  .  873  xlO6 


Copper, 1245xl06 

Platinum,  ....  1704  x  106 

Iron, 1861  xlO6 

Steel, 1955  xlO6 


The  coefficient  of  elasticity  decreases  with  increase  of  tempera- 
ture between  15°  and  200°  0. 

24.  An  isotropic  solid  has,  in  addition  to  the  modulus  of  longi- 
tudinal elasticity,  a  modulus  of  rigidity  ;  tho  former  relating  to  the 
elasticity  of  bulk  or  volume,  and  the  latter  to  that  of  shape.  If  a 
bar  be  of  square  cross-section  before  elongation,  it  will  be  found 
afterwards  to  have  undergone  deformation  in  its  angles,  although 
the  diagonals  of  the  cross-section  may  still  be  at  right  angles.  The 
numerical  ratio  of  the  intensity  of  the  force  applied,  to  the  deforma- 
tion produced  is  the  modulus  of  rigidity.  The  deformation  is 
measured  by  the  change  in  each  of  the  four  right  angles,  in  terms 
of  the  radian  (57°.  29)  as  unity. 


RELATING    TO    SOUND    AND    LIGHT. 


25 


Let 
to  the 


25.   Fundamental  Coefficients  of  Elasticity. 

there  be  a  rectangular  parallelopipedon  AH,  subjected  at  first 

action    of    equal    and    opposite    normal 

pressures  on  the  two  bases  AD  and  EH. 

The  vertical   edges  will,  by   the  laws  of 

elongation,  shorten,   and   the   horizontal 

edges  increase  in  length ;  and  the  relative 

changes  in  length  will  be  proportional  to 

the  quotient  of  the  normal  pressures  by 

the  area  AD ;  that  is,  to  the  pressure  on 

the  unit  of  area. 

Let  «  be  the  relative  shortening  of  the  Figure  2, 

vertical  edges,  ft  the  relative  increase  of 
the  horizontal  edges,  and  P  the  pressure  on  the  unit  of  area,  then 


a  —  mP, 


=  nP, 


m  and  n  being  coefficients  to  be  determined  only  by  experiment. 
If  Q  be  the  pressure  applied  to  the  unit  area  on  the  faces  AF  and 
CH,  the  edge  AC  will  be  shortened  «',  and  the  edges  AB,  AE 
lengthened  /3',  and  we  will  have 

a'  =  mQ,  f3'  =  nQ. 

If  R  be  the  pressure  on  the  unit  area  of  the  faces  AG  and  BH, 
the  edge  AB  will  be  shortened  «",  and  the  edges  AE  and  AC  elon- 
gated /3",  and  we  will  have 

a"  =  mR,  ft"  =  nR. 

If  now  the  three  pairs  of  pressure,  P,  Q,  7?,  act  simultaneously, 
their  effects  will  be  superposed,  and,  representing  by  e,  e',  e",  the 
relative  variations  of  the  lengths  of  the  edges  AE,  AC,  and  AB,  we 
will  have 

e    =a   -_(0'  +  0")  =  mP-n(Q  +  R),   ) 

e'  =  a'  -  (ft  +  ft")  =  mQ-n(P  +  R),   \  (2) 

B"  =  «"-(0  +0')  =  mR-n(P+Q)i  ) 

from  which  we  readily  deduce, 

P  =  He    +K(e'  +  e"),  ) 

Q  =  He'   +K(e-+e"),  V  (3) 

R  =  He"  +  Ke  +  e'      ) 


26  ELEMENTS    OF    WAVE    MOTION. 


,  .  ,  rr  m  —  n 

in  which  H  = 


m  (m  —  n)  — 


K  =  n I 

m  (m  —  n)  —  2n2   J 


Hence  the  pressures  exerted  upon  the  faces  of  the  volume,  and 
therefore  the  elastic  reactions,  can  be  expressed  as  linear  functions 
of  the  relative  variations  of  the  length  of  the  edges  by  means  of  two 
constant  coefficients.  These  two  coefficients,  H  and  K,  are  funda- 
mental in  the  theory  of  elasticity.  They  can  only  be  determined 
by  experimental  investigations  ;  -once  determined  for  any  body,  the 
problems  of  elasticity  become  those  of  rational  mechanics. 

Exact  analysis  of  the  conditions  of  equilibrium  in  the  interior 
of  a  solid  elastic  body  shows  that,  in  each  point  of  the  body,  there 
exist  three  rectangular  directions,  variable  from  one  point  to  an- 
other, such  that  the  elements  perpendicular  to  these  directions 
support  normal  pressures  or  tractions. 

An  infinitely  small  parallelopipedon,  having  its  edges  parallel  to 
these  three  directions,  is  in  the  condition  of  that  discussed  above; 
and  it  suffices  to  express,  in  a  general  manner,  the  relations  which 
exist  between  the  pressures  which  it  sustains  and  the  changes  of 
length  of  its  infinitely  small  dimensions,  to  obtain  the  differential 
•equations  of  the  problem  under  consideration. 

26.   Equations  (3)  can  be  written, 

p  =  (H-  K)e    +  K(e  +  e'  +  e"),  } 

Q  =  (H  —  K)  e'   +  K(B  +  e'  +  e"),  [•  (5) 

R  =  (H—  K)  e"  +  K(e  +  e'  +  e").  J 

Calling  0  the  relative  variation  of  the  volume,  or  cubic  dilata- 
tion, we  may,  because  of  the  small  values  of  the  deformations,  write 

6  =  e  +  e'  +  e".  (6) 

Placing  H  —  K  =  2/*  and  K  =  A,  we  have 

Q  =  A0  +  tyie',    V  (7) 

Each  of  the  tractions  or  pressures  is  then  the  sum  of  a  term  pro- 
portional to  the  cubic  dilatation  and  of  a  term  proportional  to  the 
linear  dilatation  parallel  to  the  pressure  considered. 


RELATING    TO    SOUND    AND    LIGHT.  27 

27.  A  liquid  parallelepiped  on  can  be  in  equilibrium  only  when 
the  pressures  exerted  on  its  six  faces  are  equal  ;  and  we  know  be- 
sides that  the  increase  of  density  or  negative  increase  of  volume  of 
the  liquid  is  proportional  to  the  pressure.     We  will  then  have 

P  =  Q  =  R  =  M.  (8) 

The  same  general  theory  thus  comprises  both  liquids  and  solids, 
in  admitting  the  coefficient  2{i  of  the  former  to  be  zero.  The  varia- 
tion of  this  coefficient  from  zero  marks  the  departure  of  the  body 
from  the  perfect  liquid  state  and  its  approach  to  that  of  the  solid. 

28.  Analytical  expression  of  the  elastic  forces  developed 
in  the  motion  of  a  system  of  molecules,  solicited  by  the  forces 
of  attraction  or  repulsion,  and  subjected  to  small  displace- 
ments from  their  positions  of  equilibrium. 


Let  x,  y,  z,  and  #  +  A#,  y  +  ky,  z-fAz,  be  the  rectangular  co- 
ordinates of  the  two  molecules  of  the  system,  whose  masses  are 
respectively  m  and  ^  and  whose  distance  apart  is  r.  The  intensity 
of  the  reciprocal  action  of  the  molecules,  being  exerted  along  the 
right  line  joining  them,  is 


f(r)  being  an  undetermined  function  of  the  distance.     If  the  sys- 
tem is  in  equilibrium,  we  have  the  relations, 


(9) 

Az 


At  a  certain  instant,  let  us  suppose  that  the  molecules  of  the 
system  are  displaced  from  their  positions  of  equilibrium  by  a  very 
small  distance,  and  let  £,  T/,  £,  be  the  projections  of  the  displace- 
ment e  of  the  molecule  m  on  the  axes;  let  £-f-A&  77  +  AT/,  £+A£, 
be  the  projections  of  the  displacement  of  the  molecule  fi  on  the 
axes;  and  r-f  p  the  new  distance  between  the  molecules.  Kepre- 
senting  the  components  of  the  elastic  force  parallel  to  the  axes 
exerted  upon  the  molecule  m  by  all  the  molecules  fi  within  the 


28 


ELEMENTS    OF    WAVE    MOTION. 


sphere  of  molecular  activity,  by  Xe,  Ye,  Ze,  so  that  JT",  Y.  Z,  are 
the  components  of  the  elastic  force  for  a  displacement  unity  in  the 
same  directions,  we  have 


Xe  = 


(10) 


29.  Developing  f(r  +  p),  and  neglecting  the  terms  of  a  higher 
order  than  those  containing  p,  since  the  displacements  are  regarded 
as  very  small,  we  obtain,  recollecting  that  A|,  A??,  A£,  are  of  the 
same  order  of  magnitude  as  p,  while  A#,  A?/,  Az,  may  be  of  any  order 
whatever, 


X,  = 


we  also  have  r2  =  Az2  +  Ay2  +  Az2, 

A#  A£  +  Aty  A??  +  Az  A<T 

from  which  p  =  -     — J ^—  —  • 

r 

Substituting  this  value  of  p  in  equations  (11),  we  obtain 


(11) 

(12) 
(13) 

(14) 


/(r)1  As  Ay 

r  r2 


(15) 


RELATING    TO    SOUND    AND    LIGHT. 

Similarly,  for  the  axes  Y  and  Z  we  get, 


+  •, 


(16) 


(17) 


Putting  0  (r)  for  V  M  for  f'(r)  -         -  ;    and  m      , 

-JL  mT~,  for  their  equals  Xe9  Ye,  Ze,  we  have 
dt         ctt 


=  m  y  =  m  ^  j  [ 


(r)  +  i>  (r)          A? 


which  give  the  values  of  the  component  elastic  forces  developed  in 
any  molecule  of  the  medium,  when  the  displacements  are  small. 


30 


ELEMENTS    OF    WAVE    MOTION. 


30.   If  the  displacement  is  only  in  the  direction  of  each  axis  i» 
succession,  we  have  the  following  groups  of  equations. 


Of*: 


r)  +  V  (r)  A?, 


Ofy: 


F2  =  m 


(20) 


Of  «: 


=±  m 


-    A?, 


(21) 


31.   Combining  the  above  equations,  we  have 


=  r,  +  F,  + 


(22) 


From  Eqs.  (19)  we  see  that  the  total  intensity 
of  the  elastic  force  developed  is  proportional  to  the  relative  displace- 
ment A£,  and  since  the  axis  has  been  assumed  arbitrarily,  it  can  be 
said,  in  general,  that  the  total  intensity,  e\/X2  -f  F2  +  ^2,  devel- 
oped, is  directly  proportional  to  the  general  relative  displacement, 


RELATING    TO    SOUND    AND    LIGHT. 


31 


From  Eqs.  (22)  we  conclude  that  the  component  intensity  of  the 
elastic  force  developed  in  the  direction  of  any  axis,  due  to  any  dis- 
placement, is  equal  to  the  sum  of  the  three  component  intensities 
developed  by  three  successive  displacements  along  these  axes,  equal 
to  the  respective  projections  of  the  general  displacement  on  these 
axes. 

32.  Of  the  nine  coefficients  of  A£  A??,  A£,  given  in  Eqs.  (18), 
BIX  only  are  distinct.  Representing  these  by 


C  = 


4>(r) 


(23) 


we  can  write  Eqs.  (22), 

Xe  = 
Ye  = 
Ze  = 


B  AT?  +  D  A£ 


(24) 


from  which  we  conclude  that  the  component  elastic  force  developed 
along  any  axis,  x,  ior  example,  by  a  displacement  e  along  any  other 
axis  y  is  equal  to  the  component  elastic  force  developed  along  the 
axis  y  by  an  equal  displacement  along  the  axis  x. 

33.  From  Eqs.  (19-21)  we  see  that  when  a  displacement  is  made 
in  any  direction,  the  resulting  elastic  force  is  not,  in  general,  in  the 
same  direction.  To  find  whether  we  can  refer  the  system  to  rectan- 
gular co-ordinate  axes,  so  that  when  a  displacement  is  made  along 
such  an  axis,  exceptional  elastic  forces  will  be  developed,  whose 
total  resultant  will  be  in  the  direction  of  the  displacement,  let «,  ft,  y, 


32 


ELEMENTS    OF    WAVE    MOTION. 


be  the  angles  which  the  direction  of  the  displacement  makes  with 
the  axes;  A,  //,  and  v,  the  angles  which  the  resultant  elastic  force 
makes  with  the  same  axes ;  then  we  have 


cos  A  = 


cos  t  = 


cos  v  = 


cos  «  =  — — 


X 


+  r2  +  z* 

Y 


+ 


+ 


cos  (3  = 


+  Af  + 
AT? 


+ 


cos  y  =  — — 


(25) 


Since  the  resultant  intensity  of  the  elastic  force  is  proportional 
to  the  displacement,  we  may  let  K  represent  the  intensity  of  the 
elastic  force  corresponding  to  a  displacement  equal  to  unity. 
K  varying  with  the  direction  of  the  displacement,  we  can  then 
place 


AT?2  +  AC2  for  its  representative,  e  V^T2+  Y*+Z\ 
and  the  first  of  Eqs.  (25)  will  become 

Xe 


cos  /t  = 


COS  [I  = 


cos  v  = 


Ye 


+ 


(26) 


Substituting  the  values  of  X,  Y,  Z,  A£  A??,  AC,  derived  from 
these  equations  in  Eqs.  (24),  after  omitting  the  common  factor  e, 
we  have 

JTcos  A  =  A  cos  a  +  ^  cos  f3  +  Fcosy,  } 

Kcosp  =  E  cos  a  +  B  cos  0  +  Z>  cos  y,  >  (27) 

^  cos  v  =  ^  cos  «  +  Z>  cos  j8  +  (7  cos  y.  ) 


RELATING    TO    SOUND    AND    LIGHT.  33 


Applying  the  conditions 

A  =  «,         /*  = 
we  have  the  equations  of  condition, 


(A  —  K)  cos  <*  +  ^  cos  0  +  .Pcos  y  =  0,  \ 
E  cos  a  +  (5  —  .AT)  cos  0  +  Z>  cos  y  =  0,  >  (28) 

«  +  D  cos  j3  +  ((7—  A")  cos  y  =  0;  ) 


together  with         cos2  a  +  cos2  0  -f  cos2  y  =  1,  (29) 

which  make  four  equations  containing  the  four  unknown  quantities 
a,  (3,  y,  and  jf. 

34.   In  order  that  Eqs.  (28)  may  be  true  for  the  same  set  of 
values  of  cos  a,  cos  ft,  cos  y,  we  must  have  the  determinant 


(30) 


Multiplying  Eqs.  (28)  respectively  by  D,  F,  and  E,  we  get 

(AD  —  KD)  cos  a  +  DE  cos  ft  +  Z>^cos  y  =  0,  j 

<*  +  (#.F—  FK)  cos  0  +  Z^cos  y  =  0,  >      (31) 
a  +  Z)^  cos  ft  +  (C#  —  ^^)  cos  y  =  0.  ) 

Placing  AD  —  EF  =  aD,    } 

BF—  DE  =  a'F,  (32) 

CE  -  DF  =  a"E,  ) 
we  have 

(a—  JT)Z>cos«  -f  JHFcosa  +  DEcosft  +  DFcosy  =  0,  j 
(0f—  JT)J?ooBj3  +  EFcosa  H-  DEcosft  +  DFcosy  =  0,  1(33) 
(a;/—  JE")  j^cosy  +  EFcoscc  -{-  DEcosft  +  DFcosy  =  0;  j 

from  which     (a  —  K)D  cos  «  =  (a'  —  ^T)  ^  cos  /3  }     ,    , 

=  (a"  -^)^  cosy  =  P;  j    (    } 


whence,  Cos  «  =  ^— ^ 


(35) 


34:  ELEMENTS    OF    WAVE    MOTION. 

Substituting  these  values  in  the  first  of  Eqs.  33,  we  obtain 
EF  DE  DP 


Clearing  of  fractions,  we  have 


- 


-£2^2  (K-a")  (K-a)      =  0.    (37) 
-D*F*  (K-a)  (K-a')  } 

If  DEF  be  positive,  supposing 

1°,  that  a  <  a'  <C  a",  by  substituting  for  Kt  in  succession  in 
Eq.  (37),  —  oo  ,  a,  a',  a",   -f  <x>  ,  we  obtain 

—  00, 

-E*F*(a'  -a)  (a!'  —  a), 
a"  -a')  (a1  -a), 
a"  -a)  (a"  -a1), 


which,  since  there  are  three  variations  in  the  signs,  shows  that 
Eq.  (37)  has  three  real  roots,  one  lying  between  a  and  a',  one  be- 
tween a'  and  a",  and  the  third  between  a"  and  oo  .  Similarly,  if 
DEF  be  negative,  the  real  roots  will  be  found  as  above. 

2°.   If  two  of  the  quantities  «,  a',  a",  are  equal,  as,  for  example, 
a'  r~.  a",  Eq.  (37)  reduces  to 

(K-  a')  \[EF(K-  a')]  {D(K-  a)  -  EF} 

~ 


which  gives  a  real  root  between  a  and  a',  a  second  equal  to  a',  and 
a  third  greater  than  a'. 

3°.  If  the  three  quantities  a,  a',  a",  are  equal,  Eq.  (37)  reduces  to 
(  K  -  a^  [DEF  (K-a)-  E*F*  -  D*E*  -  D*FZ]  =  0,    (39) 

giving  two  real  roots,  each  equal  to  a,  and  one  greater  than  a. 
Each  of  these  real  roots  of  K,  being  substituted  in  one  of  Eqs.  (35), 
will  enable  us  to  find  values  for  each  of  the  cosines  between  -f-  1 
and  —1,  and  hence  a  given  direction  for  each  value  of  K,  or  in  all 
three  directions. 


RELATING    TO    SOUND    AND    LIGHT.  '  35 

35.  We  therefore  conclude,  that  the  total  elastic  force  de- 
veloped by  any  displacement  is  not  in  general  in  the  line  of 
direction  of  the  displacement,  bat  oblique  to  it ;  that  there 
are  three  directions  at  right  angles  to  each  other,  and,  in 
general,  only  three,  along  which,  if  the  displacement  be 
made,  the  resultant  elastic  force  developed  will  be  in  the 
direction  of  the  displacement. 

36.  These  three  directions  are  called  principal  axes.     They  are 
not  specific  lines  in  a  body,  but  simply  mark  directions  along  which 
the  above  property  exists. 

37.  The  angle  which  the  direction  of  the  displacement  and  the 
resultant  elastic  force  make  with  each  other  is  given  by 

cos  U  = 


and,  if  the  displacement  be  equal  to  unity,  we  have 

U=  Xti 


=  X  cos  a  +  Y  cos  j3  +  Z  cos  y.  ) 

38.  Surfaces  of  Elasticity.    If  now  distances  which  are 
proportional  to  the  elastic  forces  developed  by  a  constant  displace- 
ment, equal  to  unity,  for  example,  in  each  direction,  be  laid  off  in 
all  directions  from  any  point  of  the  medium,  the  extremities  of 
these  lines  will  form  a  surface  which  may  be  called  a  surface  of 
elasticity.     But,  as  for  each  direction  there  are  two  things  to  con- 
sider, viz.,  the  intensity  of  the  elastic  force  and  the  angle  which  its 
direction  makes  with  the  displacement,  we  cannot,  in  general,  con- 
struct a  surface  which  would  unite  these  two  particulars. 

39.  It  will  be  shown,  hereafter,  upon  what  grounds  we  can  dis- 
regard, in   optics,    that  component  of  the  elastic  force,  7Tsin  U9 
which  is  perpendicular  to  the  displacement,  and  consider,  as  alone 
effective,  the  component  whose  intensity  is  represented  by  K  cos  U9 
parallel  to  the  displacement. 

40.  Assuming  then,  for  the  present,  that  the  effective  elastic 
force  caused  by  a  displacement  equal  to  unity  is  given  by  Eq.  (41), 
and  substituting  the  radius  vector  r  for  the  first  member,  and  the 


36  ELEMENTS    OF    WAVE    MOTION. 

values  of  JT,  Y,  Z,  from  Eqs.  (24),  and  for  A£  A??,  A£  cos  «,  cos  ft 
cos  y,  their  values  for  a  displacement  unity,  we  get 

r  =  A  cos2  a  4-  2.Z?  cos  «  cos  /3  +  2^  cos  «  cos  y  j 

+  B  cos2  13  +  W  cos  0  cos  y  +  C  cos2  y,  )    *    ' 

the  polar  equation  of  a  surface  of  elasticity  of  the  medium. 

or     //     % 
Substituting  for  cos  a,  cos  (3,  cosy,  their  values  -,  -,  -,  and 

for  r  its  equal  V%2  +  y2  +  z2,  Eq.  (42)  becomes 


8  .       .  -2  [A&  +  %2  +  Cfe*  +  2^  +  tFxz  +  Wyz\. 

(43) 

41.   Assuming  that  the  radius  vector  is   proportional  to   the 
square  root  of  the  elastic  force,  the  equation  takes  the  form 


+  2Dyz,     (44) 

which  is  the  equation  of  Fresnel's  Surface  of  Elasticity. 

42.  By  assuming  each  radius  vector  proportional  to  the  recipro- 
cal of  the  square  root  of  the  elastic  force,  Eq.  (42)  becomes 

1  =.  Ax*  +  Bf  +  Cz*  +  2Exy  +  2Fxz  +  ZDyz,          (45) 

ivhich  is  the  equation  of  what  has  been  designated  as  the  inverse 
ellipsoid  of  elasticity,  or  the  first  ellipsoid,  and  is  called  the  ellip- 
soid E. 

43.  Surfaces  of  Elasticity  referred  to  Principal 

Axes.  Principal  axes  are  those  along  which,  if  the  displacement 
be  made,  the  resultant  elastic  forces  developed  will  be  wholly  in  the 
same  direction-  We  have  seen  that,  in  any  homogeneous  medium, 
there  are  in  general  three,  and  only  three,  such  directions.  Making 
A?/,  A^;  A£,  AC;  A£,  A??,  respectively  equal  to  zero  in  Eqs.  (24),  and 
placing  A,  B,  C,  equal  to  «2,  £2,  c2,  respectively,  we  have 


. 

E  =  F  =  D  =  0, 
and  Eq.  (44)  reduces  to 

(z2  -t-  f  +  z2)2  =  a2x*  +  %2  +  &#  •  (47) 

and  Eq.  (45)  to  «V  +  %2  +  c2z*  =  1.  (48) 


RELATING    TO    SOUND    AND    LIGHT. 


37 


FresneFs  surface  of  elasticity,  Eq.  (47),  is  of  the  fourth  order, 
its  equation  being  of  the  fourth  degree.  Figure  (3)  represents  one- 
quarter  of  the  principal  section  made  by  the  plane  ac,  turned  about 
the  axis  b  through  ,an  angle  of  90°.  Taking  the  axes  to  be 


a  =  1.53, 


=  1.32,        c  =  1.00, 


we  may,  by  Eq.   (47),  readily  construct  the  principal   sections. 
Thus,  since 

r4  =  a?x*  +  TPf  +  &z\       .'.  a2  cos2  a  +  &  cos2  (3  +  c2  cos2  y  =  r2  ; 
we  have  for  the  intersection  by  the  plane  ac,  (3  =  90°,  and 


=  «2  cos2  a  4-  c3  cos2  y  =  r 


'3 


"2 


r'  =  a  cos  «,     and    r"  =  c  cos  y  =  c  sin 


Figure  3, 


Therefore  r  is  equal  to  the  hypothenuse  of  the  right-angled  tri- 
angle on  r'  and  r" ;  hence,  describe  semicircles  on  a  and  c  ;  draw 
any  right  line  from  0,  and  lay  off  on  it  a  distance  equal  to  the 
hypothenuse  on  the  intercepts  of  the  two  circles,  and  this  will  be  a 
point  of  the  curve.  Three  such  points  are  constructed  in  tha 


38  ELEMENTS   OF    WAVE    MOTION. 

figure.  The  curve  CMA  is  the  intersection  of  FresnePs  surface 
with  ac\  the  curve  CNA  is  that  of  the  ellipsoid  whose  semi-axes 
coincide  with  and  are  equal  to  those  of  the  surface  of  Fresnel ;  00 
and  00'  are  the  traces  of  the  cyclic  planes  which  contain  the  axis  b 
of  the  surface  of  elasticity  and  of  the  ellipsoid  respectively.  The 
principal  elasticities  in  crystals  never  differ  so  much  as  those 
assumed  above,  and  therefore,  in  many  cases,  the  departure  of  the 
surface  from  the  ellipsoid  is  negligible. 


.WAVE  S. 

44.  The  elastic  forces  of  the  medium,  developed  by  the  assumed 
arbitrary  displacement  of  a  molecule,  will  propagate  the  motion  in 
all  directions  from  the  point  of  initial  disturbance.  As  an  ever- 
enlarging  volume  becomes  involved  in  this  disturbance,  each  mole- 
cule takes  up  a  motion  exactly  similar  to  that  of  its  predecessor, 
which  it  transmits  in  turn  to  the  next  molecule.  This  transfer  is 
complete  when  a  single  pulse  traverses  the  medium,  and  is  both 
complete  and  continuous  when  these  pulses  are  successively  con- 
tinuous. 

In  this  latter  case  the  exciting  cause  acts  for  a  definite  portion 
of  time.  Representing  by  a  series  of  dots,  a £,  the  position  of 


Fi^dre  4, 

a  file  of  molecules  in  their  condition  of  stable  equilibrium  and  con- 
sidering alone  the  simple  case  of  rectilineal  displacements,  the 
arbitrary  displacement  of  the  molecule  m  will  give  rise  to  the  suc- 
cessive displacements  of  the  others,  and  cd  and  ef  will  represent  the 
relative  positions  of  these  molecules  at  the  end  of  a  given  subse- 
quent time  t,  equal  to  the  periodic  time  of  vibration ;  the  former, 
when  the  displacements  are  parallel  to  the  direction  of  disturbance 
propagation,  and  the  latter,  when  at  right  angles  to  this  direction. 

While,  therefore,  any  molecule  m  is  describing  its  orbit,  the  dis- 
turbance is  being  propagated  in  all  directions,  and,  at  the  instant 
the  orbit  of  m  is  completed,  the  disturbance  will  have  reached 


RELATING    TO    SOUND    AND    LIGHT.  89 

another  molecule  m',  on  the  same  line  of  direction,  which  will  then, 
for  the  first  time,  begin  to  move ;  and  the  molecules  m  and  m'  will, 
thereafter,  always  be  at  the  same  relative  distances  from  their  origins. 

45.  While  this  undulatory  motion  is  being  propagated,  mole- 
cules will  be  found  between  m  and  m',  with  all  degrees  of  displace- 
ment, both  as  to  amount  and  direction  of  motion,  consistent  with 
the  dimensions  and  shapes  of  their  orbits.     If  the  velocity  of  wave 
propagation  be  constant  in  all  directions,  the  form  assumed  by  the 
bounding  surface  containing  the  disturbed  molecules  will  be  spheri- 
cal ;  but  if  the  velocity  vary,  the  form  will  depend  upon  the  law  of 
its  variation. 

46.  This  continuous  transmission  in  any  given  direction  of  a 
relative  state  of  the  molecules,  while  the  motion  of  each  molecule 
is  orbital,  is  characteristic  of  an  undulation. 

47.  The  term  phase  is  used  to  express  the  condition  of  a  mole- 
cule with  respect  to  its  displacement  and  the  direction  of  its  motion. 
Molecules  are  said  to  be  in  similar  phases,  when  moving  in  parallel 
orbital  elements  and  in  the  same  direction;  and  in  opposite  phases, 
when  moving  in  parallel  orbital  elements  and  in  opposite  directions. 
More  generally,  similar  phases  are  those  in  which  the  anomalies  of 
the  molecule  are  the  same,  and  opposite  phases  those  in  which  the 
anomalies  differ  by  180°.     (By  anomaly  is  meant  the  angular  dis- 
tance from  an  assumed  right  line.) 

48.  A  wave  is  the  particular  form  of  aggregation  assumed  by 
the  molecules  between  the  nearest  two  consecutive  surfaces  in  which 
.similar  phases  simultaneously  exist  throughout.        t 

A  -wave  front  is  that  surface  which  contains  molecules  only  in 
the  same  phase ;  it  is  generally  understood  to  refer  to  the  surface 
upon  which  the  molecules  are  just  beginning  to  move.  The  veloc- 
ity of  a  wave  front  will  always  be  that  of  the  disturbance  propaga- 
tion. 

A  wave  length  is  the  interval,  measured  in  the  direction  of  wave 
propagation,  between  the  nearest  two  consecutive  surfaces  upon 
which  the  molecules  have  similar  phases. 

The  amplitude  of  the  undulation  is  the  maximum  displacement 
of  the  molecule  from  its  place  of  rest. 

49.  From  a  consideration  of  the  nature  of  an  undulation,  we 
.see  at  once  that,  if  A  be  the  wave  length,  r  the  periodic  time,  and 

y  the  velocity  of  wave  propagation,  we  will  have 


40  ELEMENTS    OF    WAVE    MOTION. 

-  '       '.  ;•         V=\,  (49) 

and  the  values  of  F,  A,  and  r  are  each,  theoretically,  independent 
of  the  amplitude. 

50.  To  find  an  expression  for  the  displacement  of  a  molecule  at 
any  time  during  the  transmission  of  an  undulation,  let  x  be  the  dis- 
tance of  the  molecule  from  the  origin  of  disturbance,  t  the  time 
from  the  epoch,  r  the  periodic  time  of  the  molecule,  A,  the  wave 
length,  and  V  the  velocity  of  wave  propagation.     Now,  whatever 
be  the  displacement  6  of  the  molecule  x,  at  the  time  t,  an  equal  dis- 
placement (neglecting  the  loss  due  to  increased  distance  from  the 
origin)  will  exist  for  another  molecule  at  a  distance  x  -\-  Vt' ,  at  the 
time  t  +  t'.     This  condition  gives,  whatever  be  the  value  of  t', 

d  =  <t>(x,t]  =  <t>(x  +  Vt',t  +  t').  (50) 

x  +  Vt'  is  the  distance  from  the  origin  to  the  wave  front  at  a 
time  t  subsequent  to  the  instant  at  which  it  was  at  x.  Hence  the 
molecule  x  is  behind  the  wave  front  a  distance  Vt  —  x,  and  the  dis- 
placement, 0  (x,  t),  may  be  replaced  by  0  (  Vt  —  x) ;  therefore  we 
have 

d  =  4>(x,t)  =  0(F*-a;),  (51) 

as  the  form  of  the  function. 

51.  We  have  implicitly  assumed  the  medium  to  be  in  a  state  of 
stable  equilibrium  -during  the  passage  of  the  undulation,  and,  there- 
fore, the  molecule  will  necessarily  describe  a  closed  orbit  about  its 
place  of  relative  rest.     This  orbit  may,  from  the  circumstances  of 
the  case,  be  of  the  most  varied  character,  and,  after  the  energy  due 
to  the  disturbance  has  been  dissipated,  the  molecule  will  resume  its 
original  place  of  relative  rest,  until  again  displaced  by  some  new 
disturbance.     It  is  necessary,  in  this  discussion,  to  consider  those 
disturbances  alone  which  are  regular  and  periodic,  and  to  consider 
the  orbit  after  it  has  become  determinate.    We  therefore  limit  the 
discussion  to  that  of  the  regular  periodic  disturbance,  and  the  orbit 
to  that  of  the  ellipse  or  any  of  its  particular  cases,  such  as  the 
ellipse,  the  circle,  or  the  right  line. 

52.  Simple  Harmonic  Motion.    If  a  point  a  (Fig.  5) 
move  uniformly  in  a  circular  orbit,  tlie  distance  of  its  projection 


JFTC 


RELATING    TO    SOUND    AND    LIGHT. 


41 


Tom  the  centre,  upon  the  vertical  diameter,  can  always  be  found 

From  the  equation 

/o~/         \ 

(52) 


Figure  5. 


y  —  a  sin  ^—  +  «J, 


in  which  y  is  the  required  displacement  at  the  time  t,  a  is  the  am- 
plitude or  maximum  displacement,  r  the  periodic  time,  and  a  the 
angle  included  between  the  horizontal  diam- 
eter and  that  passing  through  the  origin  of 
motion. 

2rrt 
The  angle  --  -  +  «  is  called  the  phase  of 

the  vibration,  and  may  be  made  of  any  value 
by  changing  the  arbitrary  arc  #,  the  time  t, 
or  both  together.  The  same  value  will  apply 
to  motion  along  any  diameter.  Such  mo- 
tions are  called  simple  harmonic  motions. 
It  may  easily  be  shown  that  any  two  simple  harmonic  motions,  in 
one  line  and  of  the  same  period,  may  be  compounded  into  a  single 
simple  harmonic  motion  of  the  same  period,  but  whose  amplitude 
is  equal  to  the  diagonal  of  a  parallelogram  constructed  on  the  am- 
plitudes of  the  components  inclined  to  each  other  by  an  angle  equal 
to  their  difference  of  phase. 

53.  The  Harmonic  Curve.  If  the  motion  of  a  point  be 
compounded  of  a  rectilineal  harmonic  vibration,  and  of  uniform 
motion  in  a  straight  line  perpendicular  to  the  vibration,  the  point 
will  describe  a  plane  curve,  which  is  called  the  harmonic  curve. 

Let  the  vibration  be  along  the  axis  of  y,  and  uniform  motion 
along  the  axis  x ;  we  will  then  have 


for  the  ordinates,  and 


.     font 

y  =  a  sm  ( — - 

x  =  Vt 


(53) 
(54) 


for  the  abscissas,  due  to  the  uniform  motion.  Combining  these 
equations,  eliminating  #,  and  replacing  VT  by  its  equal  A,  Eq.  (49), 
page  40,  we  have,  for  the  equation  of  the  harmonic  curve, 


y  =  a  sfn  (-^-  +  a) ; 

\    A  / 


(55) 


42  ELEMENTS    OF    WAVE    MOTION. 

in  which  A.  is  the  wave  length.  Substituting  for  x,  x  ±  i^,  the 
value  of  y  remains  the  same  for  all  integral  values  of  i.  The  curve, 
therefore,  consists  of  an  infinite  number  of  similar  parts,  which  are 
symmetrical  with  respect  to  the  axis  of  x. 


Figure  6, 

54.  To  construct  the  curve  by  points,  divide  the  circumference 
into  any  number,  as  twelve,  equal  parts ;  lay  off  on  the  axis  of  ab- 
scissas twelve  equal  distances,  corresponding  to  the  positions  of  the 
point  in  uniform  motion,  erect  ordinates  at  these  points  and  make 
them  equal  to  the  corresponding  displacements  at  the  given  times, 
and  we  have  the  curve  as  follows  : 


\6          1  &  9  10         11        18 


55.  The  varying  velocities  of  a  point  of  a  simple  pendulum  in 
motion  can  be  represented  by  the  ordinates  of  the  harmonic  curve  ; 
and  because  of  this  analogy  all  vibrations  represented  by  these 
curves  are  called  simple  or  pendular  vibrations.     The  vibration  is 
taken  to  be  the  complete  oscillation,  from  the  time  at  which  the 
moving  point  was  in  one  position  until  it  returns  to  the  same  posi- 
tion again.     By  this  definition,  the  duration  of  the  vibration  of  a 
second's  pendulum  would  be  two  seconds,  and  not  one  second. 

56.  Composition  of  Harmonic  Curves.    Let 

y'  =  a  sin  (~-  +  «),  (56) 

\    A.  / 

y"  =  I  sin  (^--  +  ft),  (57) 

\    A  / 

b'e  the  equations  of  any  two  harmonic  curves,  having  the  same  wave 
length,  but  different  amplitudes.    The  resultant  value  of  y  will  be 


RELATING    TO    SOUND    AND    LIGHT.  43 

y  =  c  sin  (^-  +  r),  (58) 

which  is  the  equation  of  another  harmonic  curve,  of  equal  wave 
length,  but  of  different  amplitude  from  either  of  the  components. 
The  values  of  c  and  y  are  given  by 

c  cos  y  =  a  cos  a  -f  b  cos  0,  (59) 

c  sin  y  =  a  sin  a  -f  £  sin  0,  (60) 


c  =      a*  +  0»  +  2a£  cos  (a  —  0).  (61) 

From  the  last  equation  we  see  that  c  may  have  any  value  be- 
tween the  sum  and  difference  of  a  and  Z>,  depending  upon  the  value 
of  the  difference  of  phase,  a  —  0,  of  the  components. 

By  a  similar  process,  it  can  be  shown  that  any  number  of  com- 
ponent harmonic  curves,  of  the  same  wave  length,  may  be  com- 
pounded into  a  single  resultant  harmonic  curve  having  an  equal 
wave  length,  but  whose  amplitude  and  phase  differ  in  general  from 
those  of  any  of  its  components. 

57.  If  the  component  curves  have  different  wave  lengths,  they 
oannot  be  compounded  into  a  single  harmonic  curve  ;  but  when 
their  wave  lengths  are  commensurable,  they  can  be  compounded 
into  a  periodic  curve,  whose  period  is  the  least  common  multiple  of 
their  several  periods.  Thus,  in  the  first  case,  where  the  wave  lengths 
are  unequal  and  incommensurable  for  the  resultant  ordinate, 


/%TTX          \  /2nx       J\  /ZTTX         \    , 

y  =  a  sin  1-y,-  +  a)  +  I  sin  i-^r  +  0)  +  c  sin  h™  +  y  I  +...., 

(62) 
in  which  the  period  is  infinite,  or  the  curve  is  non-periodic. 

In  the  second  case,  let 


m,  n,  r,  being  integers  ;  then  the  above  equation  becomes 


/2nrx 
y  =  asn—  _  +  a  .Sm—  r--  c  sml-y- 

(64) 
which,  although  not  admitting  of  reduction  to  a  simpler  form,  gives 


44  ELEMENTS    OF    WAVE    MOTION. 

constantly  recurring,  values  of  y  when  for  x  we  substitute  x  -f-  A. 
The  wave  length  of  the  resultant  curve  is  therefore  A,  and  the  curve 
is  periodic. 

58.  The  forms  of  the  component  curves  depend  only  upon  the 
wave  lengths  and  amplitudes ;  but  their  positions  on  the  axis  de- 
pend on  the  values  of  the  phase  «,  (3,  y,  etc.     By  assigning  arbitrary 
values  to  these,  we  may  shift  any  curve  along  the  axis  any  desired 
part  of  its  wave  length.     Any  such  shifting  for  any  one  or  more  of 
the  component  curves  will  necessarily  alter  the  form  of  the  result- 
ant curve,  but  will  not  change  its  wave  length. 

59.  If  the  wave  length  of  the  resultant  curve  be  assumed,  the 
wave  lengths  of  its  components  may  be  all  possible  aliquot  parts  of 
A,  and  the  number  of  the  possible  components  is  therefore  unlimited. 
Therefore  every  possible  curve  of  wave  length  A,  which  could  be  so 
constructed  from  such  component  curves,  would  be  found  among 
those  produced  by  placing,  along  the  same  axis,  an  unlimited  num- 
ber of  harmonic  curves,  as  components,  with  wave  lengths  A,  -JA? 
|A,  etc.,    . . . 

By  varying  the  amplitudes  of  the  components  and  shifting  them 
arbitrarily  along  the  axis,  an  infinite  number  of  resultants  can  be 
produced,  all  having  the  same  wave  length  A.  Fourier's  theorem 
demonstrates  that  every  possible  variety  of  periodic  curve,  of  given 
wave  length  A,  can  be  so  produced,  provided  that  the  ordinate  is 
always  finite  and  that  the  moving  point  is  assumed  to  move  always 
in  the  same  direction. 

60.  A  periodic  series  is  one  whose  terms  contain  sines  or  cosines 
of  the  variable,  or  of  its  multiples ;  thus, 

A  !  cos  x  +  A  2  cos  %x  -f  A  3  cos  3x  -+-  . . . .  An  cos  nx  -j- 

is  a  periodic  series.  This  series  goes  through  a  succession  of  values 
as  the  arc  increases  from  0  to  %TT  ;  for,  every  term  has  the  same 
value  at  the  end  and  at  the  beginning  of  that  period,  and  this  con- 
tinuously, so  that  whatever  n  may  be,  the  period  of  the  function 

is  2n. 

61.  Fourier's  Theorem  has  for  its  object  the  determination  of 
the   unknown   constants,    AQ,  A19  A2, , . .  .B^  B%,  B3, . .  . .,   and 
the  determination  of  the  conditions  by  which  any  given  function, 
y  =  /(#),  can  be  expressed  in  the  form  of 


RELATING    TO    SOUND    AND    LIGHT.  45 


—f(x)  =  Ao  +  Ai  cos  a  +  .48  cos 
+  B   sin  a;  +  #   sin 


) 
1 


The  non-periodic  term  J0  is  introduced  to  make  the  theorem 
conform  to  the  most  general  case.  If  the  function  is  capable  of 
expression  in  periodic  terms  only,  then  AQ  =  0;  this  fact  can  only 
be  determined  by  considering  each  special  case. 

The  equation  which  expresses  the  mathematical  statement  of 
Fourier's  Theorem  is 


y  =  y  „  +  S1,  1  :  Cl  sin  (         +  «,),  (66) 

in  which  «/0  is  the  mean  value  of  y,  and  each  of  the  variable  terms 
represents,  by  itself,  a  harmonic  vibration  of  which  the  period  is  an 
aliquot  part  of  the  whole  period  T. 

62,    Wave  Function.    Kesuming  Eq.  (51), 


we  see  that,  since  the  displacement  6  passes  through  all  of  its  values 
while  the  undulation  advances  a  distance  equal  .to  its  wave  length 
A,  it  has  the  properties  of  simple  harmonic  motion,  and,  therefore, 
may  be  written 

6  =  ccsin  ~(Vt  —  x).  (67) 

This  is  called  the  wave  function.  By  making  t  vary  continu- 
ously through  all  values  from  t  =  ^  to  t  =  —  ~-  ,  d  will  increase 
from  zero  to  +  «,  decrease  then  to  —  a,  and  finally  return  to 
zero,  during  the  time  ^,  which  is  evidently  the  interval  of  time 

required  for  the  undulation  to  pass  over  the  wave  length  A.  Again, 
supposing  t  to  remain  constant  and  x  to  vary  through  all  values 
from  Vt  —  A  to  Vt,  we  obtain  again  all  possible  values  of  the  dis- 
placement, which  values  will  evidently  belong,  at  the  same  instant, 
to  all  molecules  in  the  wave  length.  The  following  diagram  illus- 
trates the  two  cases  : 


Figure  8. 


46  ELEMENTS    OF    WAVE    MOTION. 

By  the  addition  of  an  arbitrary  arc  we  can  cause  the  displace- 
ment to  take  any  one  of  its  values,  at  any  time  t,  and  thus  change- 
our  origin  at  pleasure. 

63.  The  corresponding  expression  for  the  velocity  of  the  mole- 
cule in  its  rectilineal  orbit,  sometimes  called  the  velocity  of  the  wave 
element,  in  contradistinction  to  the  velocity  of  wave  propagation,  is 
given  by  ^n 

u  =  a  cos  -y-  (  Vt  —  x).  (68) 

x/ 

64.  The  principle  of  the  coexistence  and  superposition  of  small 
motions  is  shown  in  Mechanics  to  be  applicable  to  planetary  per- 
turbations.    It  is,  for  similar  reasons,  applicable  to  the  determina- 
tion of  the  resultant  displacement  of  a  single  molecule,  arising 
from  the  concurrent  effect  of  many  disturbing  causes  acting  sepa- 
rately.    The  acceptance  of  this  principle  is  equivalent  to  assuming* 
that  the  several  displacements  are  so  small  that  their  products  and 
powers  higher  than  the  first  are  negligible  with  respect  to  the  dis- 
placements themselves  ;   and  it  embodies  the  primary  supposition 
that  the  intensity  of  elastic  forces  developed  varies  directly  with  the 
degree  of  displacement. 

65.  Wave  Interference.     If  we  apply  this  principle  to  de- 
termine the  displacement  of  a  molecule  by  two  disturbing  causes, 
giving  rise  to  two  undulations  of  the  same  wave  length,  we  will 
have  for  the  first, 

6'  =  a'  sin  p^  (  Vt  -  x)  +  -4'1  ;  (69) 

for  the  second, 

6"  =  a"  sin  [y  (  Vt  -  x)  +  A"^  .  (70) 

The  total  displacement  will  be 
6'  +  (5"  =  (5  =  (a1  sin  A'  +  a"  sin  A")  cos  [^  (  Vt  -  fc)~] 

i    (71) 

+  («'  cos  A'  +  a"  cos  A")  sin     y  (  Vt  -  x)\, 
whicH  may  be  put  under  the  form 


=  a  sm         (Vt  -x)  +  A  (72) 


by  placing 

a  cos  A  —  a  cos  A'  +  a"  cos  A", 
a  sin  A  =  a  sin  A'  +  «"  sin  .4". 

S 

Whence, 

«2  =  c*'2  +  ec"2  4.  2«r«//  cos  (.4'  —  A 

j 

fan    /4    —   — 

RELATING    TO    SOUND    AND    LIGHT.  47 

(74) 

(75)  ' 
cos  ^     +  «   cos  - 

By  Eq.  (72)  we  see  that  the  resultant  undulation  is  of  the  same 
wave  length  as  the  components  ;  that  the  maximum  displacement 
of  the  resultant  undulation  is  not,  in  general,  equal  to  that  of  either 
of  the  components,  and  that  it  does  not  occur  at  the  same  time  nor 
place  with  either  of  them. 

66.   Taking  the  square  root  of  Eq-  (74),  we  have 


a  =      «'2  +  «"8  +  2«'«"  cos  (A'A)  ;  (76) 

from  which  it  is  seen  that,  when  A'  —  A"  =  0,  «  —  «'  +  «"; 
and,  when  A'  —  A"  =  180°,  a  =  a'  —  a".  Hence,  in  Eq.  (75), 
A  =  A'  —  A"  in  the  first  case,  and  A  =  A'  =  180°  +  A"  in 
the  second.  The  maximum  displacement,  then,  of  the  resultant 
undulation  may  vary  between  the  sum  and  difference  of  the  maxi- 
mum displacements  of  the  two  component  undulations,  depending 
upon  the  difference  of  phase. 

If,  in  the  two  component  undulations,  a'  =  a",  «  will  be  equal 
to  2«'  when  A'  =  A",  and  vary  from  this  value  to  zero  as  the  dif- 
ference of  phase  A'  —  A"  passes  from  zero  to  180°. 

Substituting,  in  the  expression  for  the  displacement,  A'  ±  180° 
for  A',  we  will  have 


«'  sin        -(Vt-x)  +  A'  +  n     =  tt'  ^vt-x±      +  A        (77) 
which  is  exactly  the  same  as 


when  for  x  we  put  x^f  -• 

6 

Therefore,  if  we  suppose  that  two  undulations  of  the  same  wave 
length,  starting  in  the  same  phase,  meet  after  travelling  over  routes 
which  differ  by  one-half  the  wave-length,  there  will  be  no  displace- 


ELEMENTS    OF    WAVE    MOTION. 


ment  of  the  molecule  at  the  place  of  meeting,  and  complete  inter- 
ference will  result. 

The  diagrams  of  Figure  9  illus- 
trate the  composition  of  two  un- 
dulations of  equal  wave  length, 
having  the  same  phase  in  the  first 
case,  and  opposite  phases  in  the 
second  and  third  cases.  In  AB, 
the  amplitude  of  the  resultant  un- 
dulation a  is  equal  to  the  sum  of 
the  amplitudes  of  the  component 
undulations,  a'  and  a" ';  in  A'B' 
and  A"B",  equal  to  the  difference 
of  the  amplitudes.  In  A"B",  the 
displacement  of  the  molecules  is 
zero,  and  the  two  components  mu- 
tually destroy  each  other's  action. 


Figure  9, 

67.  Interference  of  any  Number  of  Undulations. 

1°  CASE.    When  the  component  undulations  have   the   same 
wave  length. 

Let  6'    =«'    8m[^  (Vt  -  x)  +  A'  1, 

L  A- 


f  — -i 

-^  (Vt  —  x)  -f-  ^4"     , 
A  J 


(7g) 


r'  =  V'rfflB|  ^(F^-a;)  + 
etc.,  etc., , 

be  the  values  of  the  several  component  displacements.    By  addition 
we  have 

6'  +  6"  +  6'"  +  etc. 


^)  L 


RELATING    TO    SOUND    AND    LIGHT.  49 

The  second  member  may  be  placed  under  the  form  of 

• 

«  sin  A  cos  -r-  ( Vt  —  x)  -f  a  cos  A  sin  —  ( Vt  —  x) 

A.  A 

r?ff  f  (80) 

=  a  sin     i-  (  Vt  -  IK)  +  A     =  <J. 

From  which  we  conclude  that  the  resultant  undulation  will 
have  the  same  wave  length  as  that  of  the  components,  but  that  in 
general  the  maximum  displacement  and  the  phase  at  the  time  t  will 
be  different  from  those  of  its  components. 

68.   2°  CASE.     Component  undulations  of  different  wave  lengths. 

If  the  wave  lengths  are  different,  the  displacements  are  of  the 
form 


which  cannot  be  combined  into  a  single  circular  function  of  the 
same  form.  If  in  addition  the  wave  velocities  also  differ,  they  may 

V        V 
be  combined  if  —  —  —  •     Hence,  undulations  of  different  wave 

A,  A 

lengths  cannot  destroy  each  other,  and  the  combined  effect  of  sev- 
eral undulations  upon  a  single  molecule  will  be  equal  to  the  alge- 
braic sum  of  their  separate  effects.  If  this  sum  should  reduce  to 
zero  for  a  given  molecule,  it  will  differ  from  zero  for  the  molecules 
immediately  preceding  and  following  it. 

69.  The  Principle  of  HwygJiens.  Since  the  displace- 
ment of  any  molecule  is  the  causa  of  the  subsequent  displacement 
of  other  molecules,  we  may  regard  the  displacement  of  the  mole- 
cules upon  any  wave  front  as  the  cause  of  the  subsequent  displace- 
ment of  the  molecules  upon  any  other  front  which  the  wave 
afterwards  reaches.  We  may  therefore  consider  each  molecule  of 
the  wave  front  in  any  of  its  anterior  positions  as  being  the  origin, 
and  its  displacement  as  the  cause  of  secondary  waves,  each  of  which 
proceed  with  the  same  velocity.  The  aggregate  effect  of  all  these 
4 


50 


ELEMENTS    OF    WAVE    MOTION. 


secondary  waves  upon  any  other  molecule  beyond,  or  its  resultant 
displacement,  will  evidently  be  the  same  as  that  due  to  the  primary 
wave  itself.  This  principle  is  known  as  that  of  Huyghens,  and, 
together  with  the  principle  of  interference,  is  exceedingly  fruitful  in 
explaining  many  of  the  phenomena  of  wave  motion  in  sound  and 
light. 


Figure  10. 


Let  0  be  the  origin  of  disturbance,  and  BAG  the  great  wave  in 
any  of  its  anterior  positions  before  reaching  a  molecule  P';  let 
AP'  =  I',  AB  =  AC  =  I ;  let  dz  be  any  indefinitely  small  part  of 
the  wave  front,  and  0  the  angle  made  by  the  wave  front  with  any 
right  line  I  drawn  from  P'  to  any  point  of  the  wave  front,  at  a  dis- 
tance z  from  A ;  then 


I'  =  AP'  =  V (^  —  2lz  cos  9  +  z2)  (81) 

=  Z  —  z  cos  0  4-  |=  sin2  0  +  etc.  (82) 

=  Z  —  z  cos  0,  (83) 

for  all  points  of  BAG  near  to  A,  and  for  which  ^  is  insignificant. 

The  displacement  at  P',  due  to  the  secondary  waves  originating  in 
dz,  will  therefore  be 


,        adz    .     £TT  ,  __ 
=  T™  sin  -r  ( Vt  —  I  -f  z  cos  0). 


(84) 


Replacing  AP'  by  Z,  and  integrating,  we  have  for  the  resultant 
displacement  of  P'  due  to  the  great  wave, 


=  2<*'  =  jfdz  siu^(Vt- 


cos  6) 


(85), 


cos  6 


-fi  cos  ~  (Vt  -  I  +  z  cos  0)  ; 


RELATING    TO    SOUND    AND    LIGHT.  51 
and  between  the  limits  corresponding  to  -J-  b  and  —  I, 

a/i        .     %7rb  COS  6    .            Vt  —  I  /0  _v 

(J  =  -=  ---  -  sm  -  :  --  sin  2rr  -  -  —  .  (86) 

nl  cos  6               A                         A  v     ' 

The  maximum  displacement  is  therefore 


cos 


.  cos  0  /rtw 

sm-—  T  —  -•  (87) 


70.  1°.  The  above  value  of  the  displacement  will  vary  with  b, 
6,  A,  and  I.  When,  as  in  sound,  A  is  very  great  as  compared  with  b, 

—  ^—  "  -  will  be  so  small  that  the  arc  may  be  substituted  for  the 
sine  without  material  error,  and 

'  =  £•  (88) 

which  is  independent  of  6. 

2°.  When  A  is  much  smaller  than  b,  as  in  the  case  of  light,  we 
have,  when  cos  6  is  very  small,  and  6  therefore  differs  but  little 
from  90  D,  again 

-  =  ¥•  <89> 

At  other  points,  where  6  is  not  great  and  cos  6  not  small,  the 
resultant  displacement  becomes  equal  to  zero  when 

2nb  cos  0 

-  J-  --  =  ±  TT,       ±  2n,     ±  STT,     etc.  ; 

A  2A  3A 

that  is,  when       cos  0  =  ±  ^  ,     ±  ^,     ±  ^  ,    etc. 

The  greatest  resultant  displacement,  other  than  that  indicated 
above,  will  be  found  by  making  in  Eq.  (87), 

=±1,  :  '      (90) 

and  it  will  be  equal  to  —?—  —  -  ; 

nl  cos  B  ' 

and,  since  the  intensity  of  the  sensation  is  directly  proportional  to- 


52  ELEMENTS    OF    WAVE    MOTION. 

the  square  of  the  maximum  displacements,  we  will  have  the  rela- 
tion of  the  intensities, 

!  10 

[(91) 


272       **  <  —— 

?r2^  cos2  6        P  47r2£2  cos2  0 

71.  In  acoustics  it  will  be  shown  that  the  wave  lengths  corre- 

sponding to  audible  sounds  will  vary  from  -    —  -  =  57'  to  —  - 

/cO  40000 

=  £  of  an  inch,  and  therefore  there  will  be  no  point  exterior  to  an 
aperture  where  the  displacement  will  not  occur,  and  hence  the  cor- 
responding sound  be  heard.  In  light,  the  wave  lengths  vary 
between  .000026  and  .000017  of  an  inch,  and  there  will  be,  accord- 
ing to  the  2°  case,  alternations  of  light  and  darkness  surrounding 
the  central  line  drawn  from  the  place  of  original  disturbance  to  the 
centre  of  the  aperture.  These  zones  are  called  Huyghens*  zones, 
and  will  be  again  referred  to  in  the  subject  of  diffraction. 

72.  Diffusion  and  Decay  of  Kinetic  Energy.    The 

displacement  of  any  molecule  due  to  wave  motion  of  a  given  wave 
length  is  independent  of  the  periodic  time,  and,  since  the  orbits  of 
the  molecules  are  described  in  equal  times  when  they  arise  from  a 
given  periodic  motion,  they  will  be  directly  proportional  to  the  dis- 
placements or  any  other  homologous  lines.  The  velocities,  then, 
of  the  moving  molecules  being  represented  by  v9  their  kinetic  ener- 

gies will  be  represented  by  —•    Then,  because  these  energies  are 

</  * 

transmitted  without  appreciable  loss  from  the  molecules  of  one  sur- 
face to  those  of  another,  we  will  have  the  energies  of  the  molecules 
of  the  two  homologous  surfaces, 


.,     or  =          .,";       (92) 

A     '  A  A 

that  is,  ~  :  ^-2  :  :  r'2  :  r2,  or  varying  according  to  the  law  of 

2          2 
the  inverse  square  of  the  distance.     Similarly,  we  will  have 

6"r"  =  <5V,  (93) 

or  the  maximum  displacements  inversely  proportional  to  the  dis- 
tances to  which  the  disturbance  has  been  propagated. 


RELATING    TO    SOUND    AND    LIGHT.  53 

73.  Reflection  and  Refraction.    It  is  difficult  to  con- 
ceive, satisfactorily,  in  what  manner  the  molecules  belonging  to 
two  media  of  different  elasticity  and  density  are  arranged  with  re- 
spect to  each  other  in  or  near  the  bounding  surface  which  separates 
them.     When  they  occupy  positions  of  relative  rest,  the   elastic 
forces  must  be  mutually  counterbalanced  and  must  be  equal  to  those 
affecting  the  molecules  within  the  media.     We  may  assume  the  two 
media  to  have  different  densities  and  elasticities,  and  the  relative 
positions  of  the  molecules  near  the  separating  surface  to  be  deter- 
mined by  the  action  of  the  equilibrating  molecular  forces.     But 
when  a  disturbance  arising  in  one  of  the  media  reaches  the  surface, 
the  molecules  of  the  second  medium  must,  in  general,  have  motions 
and  displacements  different  from  those  of  the  first.     If  we  consider 
alone  the  difference  in  density  of  the  molecules  of  the  media,  we 
perceive  that  the  energy  in  the  incident  wave  will  not  be  wholly 
given  up  by  the  molecules  to  their  neighbors  in  the  new  medium. 
In  either  ca?e,  whether  the  molecules  have  greater  or  less  density, 
a  return  wave  will  originate  in  the  incident  medium,  analogous  to 
the  reflected  motion  in  the  impact  of  elastic  balls.     Again,  if  the 
elasticity  of  the  media  be  different,  the  elastic  forces  for  equal  dis- 
placements will  be  different,  and  thus  cause  a  return  wave  in  the 
incident  medium.     We  may  therefore  assume,  for  the  present,  that, 
owing  to  the  different  elasticities  or  densities,  or  both,  there  will  be, 
in  general,  a  separation  of  the  incident  wave  whenever  it  meets  a 
surface  separating  two  media  of  different  density  and  elasticity. 
The  fact  of  such  a  separation  is  experimentally  demonstrated  in  the 
phenomena  of  sound  and  light.     The  velocity  of  wave  propagation 
will  be  shown  to  be  a  function  of  the  elasticity  and  density  of  the 
medium,  and  therefore  the  waves,  in  general,  will  proceed  in  the 
two  media  with  different  velocities. 

74.  The  plane  of  incidence  is  that  plane  which  is  normal  to  the 
deviating  surface  and  to  the  wave  front. 

The  plane  of  reflection  is  normal  to  the  deviating  surface  and 
to  the  reflected  wave  front;  it  coincides  with  the  plane  of  incidence. 

The  plane  of  refraction  is  normal  to  the  refracted  wave  front  and 
to  the  deviating  surface. 

75.  Diverging,    Convert/ ins/,    and   Plane    Waves. 

When  the  energy  of  molecular  disturbance  is  distributed  among 


54  ELEMENTS    OF    WAVE    MOTION. 

the  molecules,  upon  an  increasing  wave  front,  the  wave  is  said  to 
be  diverging  ;  when  among  those  of  a  decreasing  wave  front,  a  con- 
verging wave;  and  when  among  those  of  an  unchanged  wave  front, 
a  plane  wave.  An  indefinitely  small  portion  of  the  front  of  any 
diverging  wave,  taken  at  a  correspondingly  great  distance  from  the 
origin,  may,  without  sensible  error,  be  considered  as  coinciding 
throughout  with  the  tangent  plane  to  the  wave  front,  and  consid- 
ered as  a  plane  wave.  The  molecules  of  a  plane  wave  at  any 
assumed  position  are  animated  by  equal  parallel  displacements,  and 
undergo  all  their  phases  while  the  plane  wave  advances  a  distance 
equal  to  the  wave  length,  measured  in  a  direction  perpendicular  to 
the  plane. 

76.   Differentiating  Eq.  (67),  we  have 

(Pd  47T2F2 

»•  =  •  -sr  6-  (94) 

Multiplying  both  members  by  m,  the  mass  of  the  molecule,  and 
replacing  m-r^  ^J  ^s  e(lual  U$>  the  intensity  of  the  elastic  force 
developed  by  the  displacement  d,  we  have 

(95) 


whence,  U  =  -      -f  F2.  (96) 

Hence,  when  a  plane  wave  is  propagated  without  altera- 
tion in  a  homogeneous  medium,  its  velocity  of  propagation 
is  directly  proportional  to  the  square  root  of  the  elastic  force 
developed  by  the  displacement  of  its  molecules. 

77.  Reflection  and  Refraction   of  Plane  Waves. 

Let  the  incident  plane  wave  AC  (Fig.  11)  meet  the  deviating  sur- 
face at  all  points,  in  succession,  from  A  to  B.  Let  V  and  A  be  the 
velocity  of  wave  propagation  and  the  wave  length  in  the  medium  of 
incidence,  and  V  and  A'  those  in  the  medium  of  intromittance. 
Let  AB  =  ds,  and  CB  =  Vdt.  While  the  disturbance  in  the  in- 
cident wave  is  moving  from  C  to  B,  the  disturbance  from  A  as  a 
centre  will  proceed  in  all  directions  in  the  medium  of  incidence, 


RELATING    TO    SOUND    AND    LIGHT. 


55 


and  be  found,  at  the  instant  considered,  upon  the  hemisphere  whose 
radius  is  AD  =  CB  =  Vdt,  and  in  the  medium  of  intromittance 
on  the  hemisphere  whose  radius  is  AD'  =  V  dt. 

Each  point  in  the  line  AB 
will,  in  like  manner,  become  in 
succession  a  new  centre  of  dis- 
turbance, sending  secondary 
waves  into  the  media  of  inci- 
dence and  of  iutromittance, 
whose  radii  will,  at  the  instant 
the  incident  wave  reaches  B,  be 


Figure  n, 


equal  to  V  and  V  multiplied 
by  the  interval  of  time  elapsing 
between  the  instant  of  arrival  of 
the  wave  front  at  the  centre 

considered  and  that  of  its  arrival  at  B.  The  surface  through  B, 
which  is  tangent  to  all  the  reflected  pulses,  may  be  taken  as  the 
front  of  the  reflected  wave,  for  it  will  contain  more  energy  than 
any  other  surface  of  equal  area  in  the  incident  medium.  Similarly, 
'the  surface  through  B  tangent  to  all  the  refracted  pulses  will  con- 
tain more  energy  than  any  other  of  equal  area  in  the  medium  of 
intromittance,  and  may  be  taken  as  the  front  of  the  refracted  wave 
at  this  instant.  These  surfaces  are  readily  seen  to  be  planes  ;  hence, 
denoting  the  angle  CAB  =  ABD  by  0,  and  ABD'  by  0',  we  will 
have 

ds  sin  0  =  Vdt,        ds  sin  0'  =  V  dt  ;  (97) 

from  which  we  obtain 


sn 


=  --,  sin  <f>'  =  p  sin 


(98) 


which  is  known  as  SnelPs  law  of  the 
sines ;  [i  is  called  the  index  of  refrac- 
tion. 

78.  The  angles  0  and  </>'  made  by 
the  wave  fronts  with  the  deviating  sur- 
face are,  respectively,  equal  to  the 
angles  made  by  the  normals  to  the  in- 
cident and  refracted  waves  with  the 
normal  to  the  deviating  surface,  and 


Figure  12, 


56  ELEMENTS    OF    WAVE    MOTION. 

are  called  angles  of  incidence  and  refraction.  The  angles  of  inci- 
dence and  refraction  are  measured  from  the  normal  to  the  deviating 
surface  on  the  side  of  the  medium  of  incidence  to  the  normal  of  the 
incident  wave,  and  to  that  of  the  refracted  wave  produced  back 
into  the  medium  of  incidence. 

The  angle  of  reflection  is  measured  from  the  normal  to  the  de- 
viating surface  to  the  normal  to  the  reflected  wave  front,  and  is 
therefore  negative.  In  the  reflected  wave,  since  the  velocity  of 
wave  propagation  is  unchanged,  \i  is  equal  to  unity,  and  Eq.  (98) 
becomes 

sin  0  =  —  sm  </>'.  (99) 

79.  These  principles  may,  in  ordinary  cases,  then  be  summa- 
rized as  follows : 

1°.  The  planes  of  incidence,  reflection,  and  refraction  are  coin- 
cident. 

2°.  The  sine  of  the  angle  of  incidence  is  equal  to  the  index  of 
refraction  or  of  reflection  multiplied  by  the  sine  of  the  angle  of  re- 
fraction or  of  reflection. 

The  modifications  which  take  place  in  polarized  light  will  be 
referred  to  hereafter  in  physical  optics. 

80.  We  see  from  Art.  77  that  the  reflected  and  refracted  waves 
are  plane  when  an  incident  plane 

wave  meets  a  plane  deviating  sur- 
face. It  is  evident  also,  from  the 
construction,  that  the  reflected 
rays  are  all  normal  to  a  plane  NN' 
symmetrical  with  MM'  with  refer- 
ence to  OX ;  and  that  the  incident 
and  reflected  rays  are  directed 
from  their  corresponding  planes 
towards  the  deviating  surface. 
The  refracted  rays  are  normal  to 
a  plane  ER'  on  the  same  side  of 
the  deviating  surface  as  the  incident  wave,  and  are  also  directed 
towards  that  surface. 

81.  General   Construction   of  the  Reflected   and 
Refracted  Waves.    Let  the  deviating  surface  AB  (Fig.  14)  be 
any  whatever,  and  the  rays  proceed  from  any  origin  0 ;  take,  in 


RELATING    TO    SOUND    AND    LIGHT. 


57 


the  medium  of  incidence,  any  spherical  surface  SS',  with  centre  at 
0,  as  the  incident  diverging  wave  ;  then,  from  all  points  I,  I',  I", 
etc.,  of  AB,  describe  spheres,  whose  radii  are  equal  to  the  intercepts 
of  the  rays  between  SS'  and  AB. 
If,  now,  tangent  planes  be  drawn 
to  the  deviating  surface  at  I,  I',  I", 
etc.,  and  to  the  surface  SS'  at  the 
corresponding  points  s,  s',  s",  etc., 
each  pair  of  tangent  planes  will 
determine,  by  their  intersection,  a 
right  line,  through  which  if  a 
plane  be  passed  tangent  to  the  cor- 
responding sphere  on  the  other 
side  of  the  deviating  surface,  it 
will  be  symmetrical  with  the  in- 
finitesimal surface  of  SS'  at  s  with 
respect  to  that  of  AB  at  the  point 
I;  and  similarly  for  the  other 
points.  By  continuity,  these  points 
of  tangency  may  be  considered  as 
forming  the  envelope  of  the  re- 
flected wave.  The  direction  of  the 
reflected  rays  is  found  by  joining 
these  points  with  I,  I',  I",  etc., 
and  extending  the  lines  toward  and  beyond  the  deviating  surface. 

82.  By  the  proper  modification  of  the  radii  due  to  the  value  of 
p,  the  index  of  refraction,  the  envelope  of  the  refracted  wave  and 
the  direction  of  the  refracted  rays  may  be  constructed. 

83.  Considering  the  reflected  wave  as  a  new  incident  wave,  the 
new  reflected  wave,  by  another  deviating  surface,  can  be  constructed 
by  an  application  of  the  above  principles ;  and  since  reflection  may 
be  considered  as  refraction  whose  index  is  —  1,  the  principle  may 
be  generally  stated,  that  any  number  of  reflections  and  refractions 
may  be  replaced  by  a  single  refraction  at  a  supposable  deviating 
surface  with  a  properly  modified  index  of  refraction. 

84.  Let  DEF  (Fig.  15)  be  any  incident  wave  whose  rays  are  not 
necessarily  parallel ;  MNP  any  deviating  surface.     At  some  subse- 
quent time  t  the  incident  wave  will  occupy  some  position  such  as 
ABG,  FG   being  equal  to    EB  =  DA  =  'vt.     By  the  principle 


Figure  14. 


58 


ELEMENTS    OF    WAVE    MOTION. 


established  above,  abg  will  be  the  enveloping  surface  of  the  reflected 
wave  corresponding  to  ABG-,  and  a^b^g^  that  of  the  refracted  wave, 
and  both  will  be  concurrent,  that  is,  the  phases  of  the  molecular 
motions  on  them  will  be 
similar ;    #PGr',     #NB', 
«MA'   will    be  the   re- 
flected,    and      ^jPGrj, 
Z'jNBj,  tfjMAj   the  re- 
fracted rays. 

85.  Prolong  the  con- 
secutive rays   of  either 
the  reflected  or  refract- 
ed waves,    say  the    re- 
flected wave  abg,  until 
they  meet  two  and  two ' 
they  will  be  tangent  to 
the  surface  «j3y,  which 
is  the  evolute   of   abg. 
Since  the  reflected  rays 
are  all  normal  to  abg, 
this  evolute  will  corre- 
spond to  any  other  po- 
sition  of   the   reflected  Figupe  I5| 
wave,  also.     The  surface 

of  which  a(3y  is  a  generatrix  is  in  optics  called  the  caustic  surface. 
It  is  evident  that  the  points  of  this  caustic  are  not  concurrent, 
because  their  distances,  being  equal  to  the  radii  of  curvature  of  abg 
from  the  reflected  wave,  are  themselves  unequal;  and  points,  in 
order  to  be  concurrent,  must  be  at  equal  distances  from  the  wave 
surface.  Whether  the  caustic  be  real  or  virtual,  the  displacements 
of  its  molecules  being  either  due  to  that  of  two  rays,  or  apparently 
so,  the  energy  of  the  molecules,  and  hence  the  resulting  sensation, 
will  be  greater  than  that  due  to  but  one  ray. 

86.  When  the  evolute  «j3y  is  known,  the  various  possible  posi- 
tions of  the  reflected  wave  can  readily  be  determined.     In  the  ordi- 
nary  cases   considered   in   optics,    the   surfaces   abg  are   those   of 
revolution  ;  the  caustic  is  then  also  a  surface  of  revolution.     Sup- 
pose abg  to  be  one  of  the  generatrices  of  the  reflected  wave,  consid- 
ered as  a  surface  of  revolution,  and  a(3y  to  be  its  evolute ;  then,  by 


RELATING    TO    SOUND    AND    LIGHT.  59 

the  property  of  the  evolute,  if  the  tangent  aaa'  be  caused  to  roll  on 
4»/3y,  each  point  of  this  tangent  will  describe  one  of  the  sections  of 
the  reflected  wave.  Thus,  a'b'g',  a"(3g",  and  abg  are  such  sections; 
the  second  of  these  being  of  two  nappes,  tangent  to  each  other  and 
normal  to  the  evolute  at  the  point  (3. 

87.  The  principle  that  the  rays,  after  the  wave  has  been  sub- 
jected to  any  number  of  reflections  and  refractions,  are  all  normal 
to  a  theoretically  determinable  surface,  and  consequently  to  a  series 
of  surfaces,  of  which  any  two  intercept  the  same  length  on  all  the 
rays,  is  principally  applicable  to  the  determination  of  caustic  sur- 
faces, and  to  the  formation  of  optical  images,  and  will  therefore  be 
further  discussed  in  that  branch  of  the  subject. 

88.  Utility  of  Considering  the  Propagation  of  the 
Disturbance  by  Plane  Waves.    In  a  homogeneous  medium, 
the  arbitrary  displacement  of  a  molecule  gives  rise  to  elastic  forces 
whose  intensities  depend  on  the  degree  and  the  direction  of  the 
displacements,  and  whose  directions  are  not,  in  general,  those  of  the 
displacements.     In  Art.  (35)  we  have  seen  that  the  displacements 
must  be  made  only  in  exceptional  directions,  in  order  that  the  elas- 
tic forces  varying  directly  with  the  degree  of  the  displacement 
should  be  wholly  in  those  directions.     Should  the  orbit  of  the  dis- 
placed molecule  be  curvilinear,  it  is  evident  that,  at  each  point  of 
its  path,  the  elastic  forces  developed  would  vary  both  in  direction 
and  intensity,  and  thus  the  general  problem  becomes  one  of  extreme 
intricacy. 

89.  If,  however,  it  be  possible  to  limit  the  discussion  to  that  of 
molecules  in  the  same  plane,  all  actuated  by  equal  and  parallel  dis- 
placements, the  variation  as  to  direction  of  the  elastic  forces  may, 
perhaps,  be  eliminated.     It  has  been  shown,  Art.  76,  that  when  a 
plane  wave  is  propagated  without  alteration  in  a  homogeneous  me- 
dium, the  velocity  of  propagation  is  directly  proportional  to  the 
square  root  of  the  elastic   force  developed  by  the  displacement. 
Hence  the  importance  of  deducing  from  the  general  equations  (18) 
the  corresponding  equations  applicable  to  the  vibratory  motions 
propagated  by  plane  waves. 

90.  At  the  time  t  let  r  be  the  distance  of  the  plane  wave,  in  a 
homogeneous  medium,  from  the  origin  of  co-ordinates ;  e  the  dis- 
placement of  the  molecules  whose  co-ordinates  are  x,  y,  z ;   |,  77,  £, 


60 


ELEMENTS    OF    WAVE    MOTION. 


the  projections  of  e  on  the  rectangular  co-ordinate  axes  ;  and  «,  ft  y, 
the  angles  made  by  the  displacement  with  the  axes,  respectively. 


We  then  have         e  =  6  sin  -^  (  Vt  —  r) ; 

o__ 

£  =.  6  cos  a  sin  -y-  ( Vt  —  r) ; 

A 

77  =  (5  cos  /3  sin  -r-  (Vt  —  r) ; 

£  =  6  cos  y  sin  -y-  ( F^  —  r). 


(100) 


(101) 


Let  r  +  Ar  be  the  distance  of  the  plane  at  a  subsequent  instant 
from  the  origin,  and  I,  m,  n,  the  angles  made  by  the  normal  to  the 
plane  with  the  axes,  then 

r  =  x  cos  I  +  y  cos  m  +  z  cos  n,  (102) 

Ar  =  A#  cos  I  4-  A«/  cos  m  +  Az  cos  w.  (103) 

From  Eq.  (101)  we  have 


-f-  A|  =  d  cos  «  sin  -^-  (  Vt  —  r  —  Ar) 


=  d  cos 


a  Li 


in     .(Vt-r)  cos  ~  Ar 

A 


277  .       27T 

—  cos  ~Y  (  V\  t  —  r)  sin  —  Ar    ; 
from  which,  and  similarly  for  the  axes  y  and  z9  we  have 
A£  =  6  cos  «    sin  —  (  Vt  —  r)  (cos  -y-  Ar  —  -  1  j 


•(104) 


27T ,  _,         .     .     2n  A 
—  cos  -y-  ( F#  —  r)  sin  -r-  Ar  I, 

A  A 


2rr  ,  T7  x  /        2rr  A 

AT;  =  d  cos  j3    sm  -r-  ( Fif  —  r)  ^cos  —  Ar  — 

—  cos  -T-  (Vt  —  r)  sin  —  Ar  L 

A  A, 

A£  =  d  cos  y    sin  -r-  ( F^  —  r)  (cos  -y-  Ar  —  1 ) 

A  \          A  / 

*  2?r  .  x    .     2rr 

—  cos  -v-  (  FP  —  r)  sin  —  Ar    . 

A  A  J     ., 


(105) 


RELATING    TO    SOUND    AND    LIGHT. 


61 


Substituting  these  values  in  Eqs.  (18),  and,  since  the  medium  is 
homogeneous,  the  sums  arising  from  the  substitution  of  the  second 
part  of  the  values  of  A£,  A??,  A£  and  which  are  of  the  form 


n 
0  (r)  sin  ~  Ar, 

A, 


~    sin  -     Ar, 


(106) 


,  .  A?/  Az         2n 
r)  — £-=  -  sin  -r-  Ar, 
r2  A 

,  x  A#  Aw    .     2rr 
(r)  — 3-^  sin  —  Ar, 
r  A 


. 
-2-  sm  —  Ar, 


r)  -t  sin  -     Ar, 


r  A 

all  reduce  to  zero,  because  they  are  formed  of  terms  which,  two  and 
two,  are  equal,  with  contrary  signs ;  for,  to  the  values  of  A#,  ky,  Az, 
equal,  with  contrary  signs,  correspond  values  of  Ar  which  are  also 

equal  and  have  contrary  signs.     Then,  replacing  cos  -r-  Ar  by  its 

At 

*     equal,    1  —  2  sin2  ~  Ar,    and    6  sin  -^  ( Vt  —  r)    by    its    equal    e, 

A  A 

Eqs.  (18)  become,  for  plane  waves, 


=  ~      =  cos 


-  cos  {3  Zptf)  (r)  - 

Y 

=  —  ^—  =  cos 


sin2  -  Ar  +  cos  y 

// 


A#  AZ     .         TT 

— r-  sm2  T 
r2  A 


sin2  —  Ar 

i— 
I 


4-  cos  j3  E^    0  (r)  +  i/>  (r)  -^-  I  sin2  ^  Ar 

N  Aty  Az    .      ?. 
+  cos  y  Sft  T/J  (r)  -^-3—  sm2  -  Ar, 


62  ELEMENTS    OF    WAVE    MOTION. 


4-  cos  pzp'ip  (r)  --  sin2     Ar 

4-  cos  y  Sf*  If  (r)  +  V  (r)  -^  \  sin2  £  An 

(107) 

91.   The  conditions  for  the  propagation  of  the  plane  wave  with- 
out change  are 


-  =  —  '  =  —  '- 


(108> 


cos  «       cos  j3       cos  y 
Substituting,  in  Eq.  (107),  for  Ar  its  equal, 

Az  cos  £  -f-  Ay  cos  m  -f  A2  cos  w  =  Ar,  (109) 

and  substituting  in  Eqs.  (108)  the  values  of  X^  JT19  Z^,  thus  ob- 
tained, we  will  have  two  relations  which,  with 

cos2  a  -f  cos?  0  -h  cos2  y  =  1,  (110) 

will  enable  us  to  determine  the  angles  «,  /?,  y,  which  the  displace- 
ment  should  make  with  the  axes,  in  order  that  the  propagation  of 
the  plane  wave  may  be  possible. 

92.  Because  of  the  equality  of  the  coefficients  of  cos  j3  and 
cos  a  in  the  first  and  second  of  Eqs.  (107),  and  of  cos  J3  and  cos  y 
in  the  third  and  second,  and  of  cos  a  and  cos  y  in  the  third  and 
first,  we  can,  by  substitutions  and  reductions  similar  to  those  em- 
ployed in  Art.  33,  deduce  corresponding  principles,  and  hence 
determine  that,  for  each  direction  of  the  plane  wave,  there  corre- 
spond, for  the  molecular  displacements,  three  rectangular  directions; 
such  that  the  -plane  wave  may  be  propagated  without  change,  andi 
that  these  three  directions  are  parallel  to  the  three  axes  of  an  ellip- 
soid whose  equation  is, 


RELATING    TO    SOUND    AND    LIGHT. 


2xz 


l(l>(r)  +  1>(r)^-  |sin2^Ar 

A^2 
r2  J 
A#  Ay  ^2  TT 

(r) 


=1. 


5—  sin2  T  Ar 

f"  A 

^    g  sin2  ^  Ar 


This  is  called  either  the  inverse  ellipsoid  or  the  ellipsoid  of  polar- 
ization. Having  also  the  relation  expressed  in  Eq.  (109),  we  see- 
that  the  coefficients  of  Eq.  (Ill)  depend  upon  the  angles  I,  m,  n, 
which  determine  the  direction  of  the  plane  wave,  upon  certain  con- 
stants which  define  the  constitution  of  the  medium,  and  upon  the 
wave  length.  The  velocity  of  propagation  is  inversely  proportional 
to  the  length  of  that  axis  of  the  ellipsoid  to  which  the  molecular 
displacements  are  parallel. 

93.  Relation  between  the  Velocity  of  Wave  Propa- 
gation of  Plane  Waves  and  the  Wave  Length  in 
Isotropic  Media.  All  directions  heing  identical  in  isotropia 
media,  we  will  assume  the  plane  wave  normal  to  the  axis  of  x.  We 
then  have  Ar  =  A#,  and 


and  Eqs.  (107)  reduce  to 


(113) 


€4  ELEMENTS    OF    WAVE    MOTION. 

Xl  =  cos  «  Sp     0.  (r)  +  V  M  — f-     sin2  ^  Az, 
Fj  =  cos  (3  I,fi\  0  (r)  +  V  (^)  -|-     sin3  ^  A», 

t1 — I 
0  (r)  -f  -0  (r)  —     sin2  j  Ao; : 

and  the  equation  of  the  ellipsoid  to 


?in2  —  Ao; 


and,  since  all  directions  perpendicular  to  the  axis  of  x  are  identical 
with  reference  to  the  plane  of  the  wave,  we  have  Ay  =  A«,  and 
Eq.  (114)  of  the  ellipsoid  becomes  one  of  revolution  about  the  axis 
of  x.  Whence,  we  conclude  that,  in  an  isotropic  medium,  a  plane 
wave  normal  to  a  given  direction  can  be  propagated  without  change, 
whenever  the  molecular  displacement  is  parallel  or  perpendicular  to 
this  direction.  To  any  one  direction  of  normal  propagation  in  such 
a  medium,  there  corresponds  an  infinite  number  of  waves  with 
transversal  vibrations,  having  the  same  velocity,  and  but  one  wave 
with  longitudinal  vibrations  whose  velocity  is  different  from  those 
with  transversal  vibrations. 

94.   For  the  wave  with  longitudinal  vibrations,  we  have 
a  =  0,  0  =  y  =  90°, 


=Xt=  ZP    *  (r)  +  i>  (r) 


TT 

sin* 


and,  from  Eqs.  (96)  and  (107),  we  have 

V*  =        S         (r)  +  V  (r)  -f]  sitf  \  A*.  (116) 


RELATING    TO    SOUND    AND    LIGHT. 


65 


In  Acoustics,  it  will  be  shown  that  sound  is  due  to  longitudinal 
vibrations  of  the  medium.  This  equation  will  then  be  applicable  in 
all  cases  of  sound  arising  from  such  vibrations,  and  will  be  referred 
to,  in  that  branch  of  the  subject.  In  Optics,  it  will  be  shown  that 
transversal  vibrations  only  are  efficacious  in  producing  light. 

95.  For  waves  with  transversal  vibrations  in  isotropic  media, 
the  velocity  is  independent  of  the  direction  of  the  displacement 
We  can  then  suppose  the  displacement  parallel  to  the  axis  of  y,  ana 
thus  have 

a  =  y  =  90°,  0  =  0, 


rt  = 


and       F2 


A2    r  A 

=  —  SM  10  (r)  +  V  (r)  -£-  I  sin2  £  Aas. 


(117) 


(118) 


This  equation  is  applicable  in  light,  for  the  determination  of 
wave  velocity  in  isotropic  and  homogeneous  media,  and  will  be  used 
hereafter  in  determining  the  velocity  of  light  propagation. 

By  developing  sin2  -r  A#  into  a  series,  we  find 

A, 


Substituting  this  in  Eq.  (118),  we  obtain 


72  = 


* 


(119) 


in  which  a,  b,  c,  ----  have  for  values, 


a= 


c  = 
rf=  - 


(120) 


ELEMENTS    OF    WAVE    MOTION. 

These  constants  depend  only  on  the  constitution  of  the  medium, 
and  decrease  very  rapidly  in  value,  for  Az  is  always  a  very  small 
quantity.  If  the  wave  length  be  not  excessively  small,  if  it  sur- 
passes a  certain  value  which  observation  only  can  determine,  the 
terms  of  the  second  member  of  Eq.  (119)  will  have  very  rapidly 
decreasing  values,  and  we  will  obtain  an  expression  approximately 
near  to  V2  by  taking  only  the  first  few  terms.  Hence,  a  must  be 
positive,  and,  since  observation  shows  that  the  most  refrangible 
rays  are  those  of  the  shortest  wave  length,  and  that,  as  a  conse- 
quence, V  decreases  with  A,  b  is  necessarily  negative. 

96,  Hence,  in  isotropic  media,  the  elasticity  being  uniform  in 
all  directions,  the  form  of  the  wave  surface  will  be  spherical,  and 
when  the  displacements  are  longitudinal,  its  radius  at  the  unit  time 
from  the  epoch  will  be  the  value  of  V  obtained  from  Eq.  (116) ; 
when  the  displacements  are  transversal,  the  radius  will  be  the  value 
of  V  in  Eq.  (118).     The  former  relates  wholly  to  waves  of  sound, 
and  the  latter  to  those  of  light. 

The  subsequent  discussion  will  now  apply  to  transversal  vibra- 
tions alone,  and  the  conclusions  derived  belong  therefore  to  the 
transmission  of  light  undulations. 

Experiment  shows  that  the  media  which  transmit  the  waves  of 
light  are  not  in  general  isotropic,  and  as  a  consequence  the  form  of 
the  wave  surface  will  not  be  spherical.  We  will,  therefore,  now 
seek  the  form  of  this  surface  in  the  general  case,  and  make  use  of 
the  properties  of  plane  waves  for  this  purpose. 

97.  Plane  Waves  in  a  Homogeneous  Medium  of 
Ttiree  Unequal  Elasticities  in  Rectangular  Direc- 
tions.   In  the  plane  wave,  the  following  conclusions  have  been 
deduced : 

1°.  The  displacements  of  the  molecules,  in  each  position  of  the 
same  plane  wave,  must  be  rectilineal  and  parallel  to  each  other  and 
to  their  original  directions. 

2°.  The  elastic  forces  developed  by  these  displacements  must  be 
either  in  the  directions  of  the  displacements  or  alone  efficacious  in 
these  directions. 

3°.  The  propagation  of  the  plane  wave  unaltered  is  then 
possible. 

These  conclusions  involve,  as  consequences,  a  constancy  of  ve- 


RELATING    TO    SOUND    AND    LIGHT.  67 

locity  of  propagation  when  the  plane  wave  is  unchanged  in  direction, 
and  a  variation  in  the  velocity  as  the  direction  is  changed.  Hence, 
if  the  elasticities  of  a  homogeneous  medium  differ  in  all  directions, 
and  we  suppose  plane  waves,  having  all  possible  positions,  originate 
at  any  point  m  of  an  indefinite  medium,  these  plane  waves,  at  the 
end  of  a  unit  of  time,  will  be  at  different  distances  from  m.  The 
surface  which  is  the  envelope  of  all  these  plane  waves  at  this  instant 
is  called  the  wave  surface. 

98.  Let  a  >  I  >  c 

be  the  principal  axes  of  elasticity  of  such  a  medium.  Then  «2,  Z>2, 
c2,  will  measure  the  elastic  forces  developed  in  these  directions  by  a 
displacement  equal  to  unity,  and  any  of  the  surfaces  of  elasticity 
heretofore  determined  can  be  used  to  obtain  the  elastic  forces  devel- 
oped by  an  equal  displacement  in  the  direction  of  the  corresponding 
radius  vector  of  the  surface.  The  velocity  of  wave  propagation 
being  proportional  to  the  square  root  of  the  elastic  force,  Eq.  (96), 
its  value  can  be  found  when  the  elastic  force  due  to  the  displace- 
ment in  any  direction  is  known. 

99.  Fresnel  made  use  of  the  single-napped  surface  of  elasticity 
whose  equation  is 

&#  =  r4;  (121) 


but  for  plane  waves,  the  inverse  ellipsoid  of  elasticity  or  first 
ellipsoid, 

4-  %2  +  cW  =  1,  (E) 


together  with  its  reciprocal  ellipsoid, 
r2       ^       z2 


can  be  more  readily  used,  because  of  its  better  known  properties. 
The  squares  of  the  semi-axes  of  (W)  and  of  the  reciprocals  of  (E) 
are  the  principal  elasticities  of  the  medium. 

100.   There  are  two  cases  to  consider: 

1°.  The  plane  of  the  wave  contains  two  of  the  principal 
axes,  and  hence  is  one  of  the  principal  planes  of  the  medium. 
The  plane  cuts  the  ellipsoids  in  ellipses  whose  semi-axes  are  either 
two  of  the  principal  axes.  Whatever  be  the  direction  and  amount  of 


68  ELEMENTS    OF    WAVE    MOTION. 

the  displacement,  it  may  be  replaced  by  its  components  in  the  di- 
rection of  the  axes  proportional  to  cos  a  and  sin  «,  a  being  the 
angle  made  by  the  displacement  with  either  axis. 

Considering  these  separately,  we  see:  1°,  that  each  will  commu- 
nicate to-  the  molecules  in  the  adjacent  plane  analogous  rectilineal 
motions  which  will  be  propagated  without  alteration  of  direction; 
2°,  that  the  elasticities,  and  hence  the  velocities  of  propagation 
which  belong  to  these  two,  are  different,  and  that  after  a  time  there 
will  be  two  series  of  molecules  situated  in  parallel  planes,  parallel 
also  to  the  primitive  plane,  which  will  contain  all  of  the  original 
energy ;  3°,  that  the  vibrations  of  the  molecules  in  these  two  plane 
waves  will  be  at  right  angles  to  each  other. 

101.  2°.  The  plane  wave  is  any  whatever.  The  sections 
of  the  ellipsoids  will  be  ellipses,  but  will  not  in  general  contain 
either  of  the  axes  of  the  ellipsoids.  There  will  then  be  no  direction 
of  the  displacement  that  can  give  a  resultant  elastic  force  in  the 
direction  of  the  displacement.  It  is,  therefore,  essential  for  a  rec- 
tilineal oscillation  of  the  molecule  and  for  a  consecutive  transmis- 
sion of  this  oscillation,  that  there  should  be  no  tendency  of  the 
rectilineal  displacement  to  be  deflected  on  either  side,  but  that  the 
line  of  the  resultant  force  should  "be  projected  upon  the  displace- 
ment. As  it  is  not  in  general  in  the  plane,  but  oblique  to  it,  it  can 
fte  resolved  into  two  components  :  one  normal  to  the  plane,  which 
is  not  effective  in  light  undulations  ;  and  the  other,  which  is  alone 
•efficacious,  in  the  direction  of  the  displacement.  In  each  elliptical 
isection  there  are  two  such  directions,  which  are  named  singular 
'directions^  and  which  are  perpendicular  to  each  other.  Assume  any 
plane  section  through  the  centre  of  (E) ;  the  elasticity  measured  by 
the  squares  of  the  reciprocals  of  the  radii-vectores  is  the  same  to  the 
right  and  left  for  the  two  axes  of  the  section,  and  is  the  same  only 
for  them.  Through  either  of  the  axes  pass  the  normal  plane  to  the 
section ;  it  will  cut  all  the  parallel  plane  sections  in  their  homolo- 
gous axes.  With  reference  to  this  normal  plane,  the  radii-vectores, 
.and  therefore  the  elasticities  of  each  section,  are  symmetrical. 
Hence,  if  the  displacement  be  along  one  of  the  axes  of  the  section, 
the  total  elastic  force  will  be  in  the  normal  plane,  and  will  be  pro- 
jected on  the  axis  of  the '  section.  And  since  the  ellipsoid  semi- 
diameters  are  inversely  as  the  velocities  of  propagation,  the  recipro- 


RELATING    TO    SOUND    AND    LIGHT  69 

cals  of  the  axes  will  measure  the  velocities  of  wave  propagation. 
Hence  is  established  the  fact  that  for  each  section  there  are  two  of 
these  singular  directions,  and  that  they  are  rectangular.  These  two 
singular  directions  perform  the  same  function  for  the  vibrations  of 
the  plane  wave  as  do  the  axes  of  elasticity  themselves  when  the 
plane  wave  contains  them.  Each  vibration  is  replaced  by  two  others 
in  the  direction  of  the  singular  directions,  and  these  two  compo- 
nents proceed  in  the  medium  without  change  of  direction,  but  with 
different  velocities,  so  that  there  are  then,  in  the  general  case,  two 
plane  waves  parallel  to  each  other  and  to  the  original  plane  wave. 
If  a  be  the  angle  made  by  the  displacement  with  one  of  the  axes, 
the  component  displacements  will  be  proportional  to  cos  «  and 
sin  «,  and  the  elastic  intensities  to  cos2  cc  and  sin2  «.  Whatever 
may  be  the  direction  of  the  original  supposed  vibration  in  the  plane 
wave,  the  two  plane  waves  which  replace  it  are  always  the  two  above 
designated. 

102.  If  the  plane  of  the  wave  coincides  with  either  of  the  circu- 
lar sections  of  the  ellipsoid,  the  plane  wave  will  be  propagated 
without  alteration,  whatever  be  the  direction  of  the  displacement, 
with  a  velocity  equal  to  #,  the  reciprocal  of  the  mean  semi-axis  of 
the  ellipsoid. 

103.  The   Double-Napped  Surface  of  Elasticity. 

If  through  the  centre  of  (E)  we  pass  any  plane,  and  on  the  normal 
to  the  section  at  the  centre  set  off  distances  inversely  proportional 
to  the  semi-axes  of  the  section,  the  locus  of  all  these  pairs  of  points 
is  called  the  double-napped  surface  of  elasticity.  For,  each  radius 
vector  measures  the  velocity  of  propagation  of  one  of  the  plane 
waves,  arising  from  a  displacement  in  the  plane  of  section,  and  the 
square  of  each  of  these  normal  velocities  is  the  measure  of  the  elas- 
tic force  developed  by  the  component  displacement  along  the  axes 
of  the  section. 

104.  If  through  each  of  the  points  so  determined  planes  be 
passed  parallel  to  the  corresponding  plane  of  section,  the  envelope 
of  all  these  planes  will  be,  by  definition,  the  wave  surface.     Hence, 
the  latter  can   be  constructed  by  points  from    this    surface    of 
elasticity. 

105.  To  get  the  polar  equation  of  the  latter  surface,  let  us  take 
for  co-ordinate  axes  the  principal  axes  of  the  medium  ;  let  Z,  m,  n^ 


70  ELEMENTS    OF    WAVE    MOTION. 

be  the  angles  made  by  the  normal  to  the  plane  wave  with  these 
axes,  x,  y,  z,  respectively ;  «,  ft  y,  those  which  OLC  of  the  axes  of 
the  ellipse  of  section  make  with  the  same  axes  ;  then  we  have 

cos  a  cos  I  +  cos  )3  cos  m  +  cos  y  cos  n  =  0.  (122) 

The  elastic  force  developed  by  a  displacement  parallel  to  the 
axis  of  section  is  projected  on  the  plane  of  the  wave  parallel  to  this 
displacement,  and  its  components  are 

X  —  a2  cos  a,     Y  =  &  cos  ft     Z  =  c2  cos  y.          (123) 

The  cosines  of  the  angles  which  this  elastic  force  makes  with  the 
axes  are  then  proportional  to  these  values.  An  auxiliary  right  line 
perpendicular  to  the  direction  of  the  elastic  force  and  to  the  dis- 
placement will  lie  in  the  plane  of  the  wave,  and  if  u,  v,  w,  be  the 
angles  which  it  makes  with  the  axes,  we  will  have 

a2  cos  a  cos  u  -\-  b2  cos  j3  cos  v  +  c2  cos  y  cos  w  —  0,  \ 

cos  a  cos  u  +  cos  (3  cos  v  +  cos  y  cos  w  =  0,  I  (124) 
cos  I  cos  u  -f  cos  m  cos  v  +  cos  n  cos  w  =  0.  ) 

Representing  the  velocity  of  propagation  of  the  plane  wave  by 
V,  we  have 

V2  =  a2  cos2  «  +  W  cos2  j3  +  c2  cos2  y.  (125) 

Combining  the  above  equations,  and  eliminating  the  quantities 
«,  ft  y,  u,  v,  tv,  we  will  have  an  equation  containing  V,  I,  m,  n, 
which  will  be  that  of  the  surface  required.  To  eliminate  u,  ?;,  w, 
we  will  make  use  of  the  method  of  indeterminate  coefficients ;  thus, 
multiply  Eqs.  (124)  by  B,  A,  and  unity,  respectively,  add  the  three 
resulting  equations,  and  from  the  conditions  for  B  and  A  that  the 
coefficients  of  cos  v  and  cos  w  shall  reduce  to  zero,  we  will  have 

(A  -f  Ba°)  cos  a  -f  cos  I   =  0,  J 

(A  +  BP)  cos  0  +  cos  m  =  0,  V  (126) 

(A  4-  Be2)  cos  y  +  cos  n  =  0.  ) 

Multiply  these  by  cos  «,  cos  3,  and  cos  y,  respectively,  add, 
and  reduce  by  Eqs.  (122),  (125) ;  we  will  have 

Q.  (127) 


RELATING    TO    SOUND    AND    LIGHT.  71 

Substitute  this  value  of  A  in  Eqs.  (126),  and  we  have 

cos  I   =  B  (  F2  —  a2)  cos  a,  } 

cos  m  =  B  (  F2  —  £2)  cos  j3,  V  (128) 

cos  w  =  ^  (  F2  —  c2)  cos  y.  ) 

From  which  we  get 
cos  /  cos  m  cos  n 


F2  — 


cos  «  cos  ]3  cos  y 


cos2  Z  cos2  m  cos2 

f™     /    T7*O  19\O     "T 


—  a2)3  T  ( F2  —  J2)2  ^  ( F3  —  c2)2  J 


•   (129) 


Replacing  cos  «,  cos  0,  cos  y,   in  Eq.  (122),  by  their  propor- 

,.       ,  ....          cos  I         cos  m         cos  n 

tional  quantities,  ,  -,      8  we  have 


wL/O  V  _  V/Vk^  ffV  _  -WVKJ  IV  ^  y    .     ^     _   V 

(IdU; 

the  polar  equation  of  the  double-napped  surface  of  elasticity,  in 
which  F  is  any  radius  vector. 

106.  The  Wave  Surface.  Through  any  point  of  the  sur- 
face of  elasticity  pass  a  plane  perpendicular  to  the  radius  vector  at 
that  point,  and  let  r,  A,  p,  v,  be  the  polar  co-ordinates  of  any  point 
of  the  plane.  The  equation  of  the  plane  will  be 

cos  I  cos  A  +  cos  m  cos  fi  -j-  cos  n  cos  v  =  —         (131) 

T 

"We  have  also,  as  equations  of  condition, 

cos2 1  4-  cos2  m  4-  cos2  n  =  1,  (132) 

cos2?  cos2  m          cos2  n  /IQQ\ 

The  wave  surface  is  the  enveloping  surface  of  the  planes  given 
by  Eq.  (131),  and  its  equation  can  be  determined  by  eliminating  F, 
Z,  ??i,  n,  and  finding  an  equation  between  r,  A,  p,  v.  To  do  this, 
differentiate  Eqs.  (131),  (132),  (133),  regarding  cos  I  and  cos  m  as 
independent  variables,  and  we  will  have 


72 


ELEMENTS    OF    WAVE    MOTION. 


,  d  cos  ft       1     d  V 

cos  A  -f  cos  v  - 


7 
cos  ?  -f  cos  n 


6?  COS  I 

d  cos  ft 

*?  COS  I 


r  d  cos  I' 


COS  Z  COS  ft      (?  COS  ft 


(134) 


72  _  £2 


d  cos 


cos2 


cos2  m 


cos2  ft 


cos  m 


cos  ft       cos  n 


^  cos  ft        1      dV 

COS  JU    -f-  COS  V  -7—        -  =  —  j —      —  . 

d  cos  m       r  d  cos  m 

d  cos  ft 

cos  w  +  cos  ft  -j —    -  =•  0, 
d  cos  m 

cos2  m  cos2  0 

(J72_#2)2  +  (F2~~ 

rfF 


(  72  _  02 


+ 


»  (135) 


i /\w    m     T         i     u  cos  n    a  cos  n        **, ,  ™ ,  ,,. 

107.  To  eliminate  -= 7,  -= , ,, ,    multi- 

d  cos  I     d  cos  m    d  cos  I    d  cos  m 

ply  Eqs.  (134)  by  1,  A,  and  —  5,  respectively,  add  the  resulting 
equations  together,  and  perform  the  same  operations  on  Eqs.  (135). 
Supposing  the  indeterminate  quantities  A  and  B  to  have  such 

d  cos  ft    d  cos  ft        6?  F 


values  as  will  make  the  coefficients  of 

-j ,  equal  to  zero,  we  will  have 

dcos  m 

cos  A  -f-  A  cos  I  =  B 
cos  \i  +  A  cos  in  =  ^ 


cos  J '  d  cos  m'  <?  cos  £*" 


cos 


7?  F 
=        K 


cos  v  +  ^4  cos  ft  =  B 
cos2  ?  cos2  m 


F2  — 

cos  m 
V*— 

COS  ft 


cos2  ft 
»  —  ^)8J\ 


(136) 


Multiply  the  first  three  of  the  above  equations  by  cos  ?,  cos  my 


RELATING    TO    SOUND    AND    LIGHT.  TS 

and  cos  n,  respectively,  add  the  resulting  equations,  and  reduce  by 
Eqs.  (131),  (132),  (133) ;  we  wiU  have 


A  +  -  =  0. 


(137) 


Squaring  the  first  three  of  Eqs.  (136),  and  adding,  we  get,  after 
reduction 


1  +  2J 

y  +  ^2  =fv> 

whence,  we  have 

A            V 

r  ' 

Y 
B  =  —  (r2  —  F2).                   (139) 

T 

Substituting  these  values 

in  Eqs.  (136),  we  obtain 

r  cos 

/I        Fcos  I                            i 

•   (140) 

r  cos 

jlfc            F  COS  7ft 

r  cos 

i          r  cos2  1 

v        F  cos  n 

C2  ~'    Y%  CZ' 

cos2  m             cos2  f& 

•«  ...  F2            [_(  F2  —  a2)2 

(F2  —  ^)2       (F2  —  c*)»J 

f    cos2  A 

COS2  jLt                 COS2  V 

The  first  three  equations  (140)  can  be  placed  under  the  form, 


cos  A  -       cos  Z   =(,2_ 


cos  \i 


,  „      ,_..„,    cos  » 
cos  u cos  m  =  (r2  —  F2)  --s — %i 

A«  \  /   At2  A55 

r*os  v 

cos  v cos  n  =  (r2  —  F2)  - 

r  /  ra  _  C2 

Adding  these,  after  multiplying  tnem 


(141) 


Adding  these,  after  multiplying  tnem  tf  cos  A,  cos  p,  cos  v,. 
respectively,  and  reducing  by  Eq.  (131),  we  have 


74:  ELEMENTS    OF    WAVE    MOTION. 


whence,  dividing  by  r2  —  F2,  we  have 

cos2  A         cosy        cos2i; 

+ 


the  polar  equation  of  the  wave  surface. 

108.   A  more  advantageous  form  for  discussion  can  be  obtained 
by  subtracting  the  identical  equation, 

1       _  COS2  X         COS2  [I         COS2  V  fiAA\ 

^  -    ~fi      I  --  ^2"  "I  --  ^T~> 
from  the  equation  above,  by  which  there  results 
a2  cos2  A       #2  cos2  i      c2  cos2  v 


109.   To  obtain  the  equation  of  the  wave  surface  in  rectangular 
co-ordinates  substitute  for  cos  A,  cos  \i,  cos  v,  and  /*,  their  equals, 

-,  -,  -,  and  A/#2  +  y2  +  z2  ',  whence,  we  have 

T       T      T 


-  P  (c2 


, 

=  0.  j  ^     > 


This  equation  being  of  the  fourth  degree,  the  surface  is  of  the 
iourth  order,  and,  as  will  be  shown  hereafter,  consists  of  two  dis- 
tinct nappes,  having  but  four  points  in  common. 

110.  If  two  of  the  velocities  become  equal,  as,  for  example, 
Jb  —  c,  the  equation  gives 

a*  +  02  +  s?  =  fl»,  (147) 

aW  +  Z>2  (y2  +  z2)  =  aW,  (148) 

which  shows  that  the  wave  surface,  under  this  supposition,  consists 
-of  a  spherical  surface  and  that  of  an  ellipsoid  of  revolution  tangent 
to  the  sphere  at  the  extremity  of  its  polar  axis. 


RELATING    TO    SOUND    AND    LIGHT.  75 

Finally,  if  the  three  principal  velocities  become  equal,  or  a  =  5 
=  c,  as  in  isotropic  media,  Eq.  (146)  becomes 

y*  +   if  +   £2   _.   ^  (U9) 

and  the  wave  surface  becomes  spherical,  as  has  been  heretofore 
shown. 

111.   Construction  of  the  Wave  Surface  by  Means 
of  the  Ellipsoid  (  W).     Let  us  suppose  that  the  ellipsoid  (W), 

x*       y2       z2 

jjl  +  5  +  31-l,  (150) 

be  cut  by  any  plane  through  its  centre,  and  that  distances  be  laid 
off  on  the  normal  equal  to  the  semi-axes  of  the  elliptical  section. 
Referring  to  the  construction  of  the  double-napped  surface  of  elas- 
ticity by  means  of  the  ellipsoid  (E), 

##  =  1,  (151) 


we  see  that  in  designating  the  polar  co-ordinates  of  the  points  con- 
structed by  the  aid  of  (W)  by  r,  k,  p,  v,  the  equation  of  their  loci 
can  be  obtained  from  the  equation  of  the  double-napped  surface, 

cos2m 


by  substituting  for  F2,  «2,  J2,  c2,  I,  m,  n,  respectively,  -2,  -2,  ^,  -, 
A,  j^,  v.     We  thus  obtain 

=  o,  (153) 


«2  COS2  A         J*  COS2  |W     ,    C2  COS2  V 

-^zr^  +  tstrp  +  TJ^T?  =  °' 

which  is  the  equation  of  the  wave  surface.  Hence,  points  of  the 
wave  surface  can  be  constructed  from  the  ellipsoid  (W)  in  precisely 
the  same  manner  as  points  of  the  surface  of  elasticity  from  the 
ellipsoid  (E),  except  that  in  the  former,  distances  equal  to  the  semi- 
axes  are  laid  off  on  the  normal,  and  in  the  latter  the  distances  are 
equal  to  the  reciprocals  of  the  semi-axes. 


76 


ELEMENTS    OF    WAVE    MOTION. 


112.  Direction  of  the  Vibration  at  any  Point  of 
the  Wave  Surface.  Let  us  consider  any  plane  wave  tangent 
to  the  wave  surface;  the  displacement  propagated  by  this  plane 
wave  makes  the  angles  a,  ft  y,  with  the  axes  ;  the  radius  vector  of 
the  wave  surface  at  the  point  of  tangency  makes  with  the  axes  the 
angles  A,  //,  v  •  therefore  the  angle  between  these  two  lines  will  de- 
termine the  required  direction. 

Eliminate  in  Eq.  (129)  the  angles  I,  m,  n,  by  means  of  the  first 
three  of  Eqs.  (140),  which,  with  the  last  of  Eqs.  (136),  will  give 


cos  A 


cos  // 


cos  j3 


cos  v 


cos  y 


/    cos2  A 


I       /     9  19\9       I 


cos2  v 


cos2  m  cos2  n 

~~^2\2  ~*~  I  172  _  A2\2  "*"  7  F2 />S 


F;    /  i_     i_ 

""   r  \  BVr  ~  ^^±72' 


(155) 


whence, 


cos  A 
^^"S 

cos  // 

COS  V 


cos  a 


cos  j3 


cos  y 


(156) 


Substituting  in  Eq.  (143)  of  the  wave  surface,  we  have 


/         F2 

cos  a  cos  A  -f-  cos  )3  cos  \i  -f-  cos  y  cos  v  =  \J  1 -•    (157) 


In  the  figure,  let  M  be  any  point  of 
the  wave  surface,  OM  the  radius  vector, 
and  OP  the  perpendicular  F  on  the  tan- 

F  o 

gent  plane  to  the  wave  surface ;  -  -  is 


Figure  16. 


RELATING    TO    SOUND    AND    LIGHT.  77 

/           F* 
then  the  cosine  of  POM,  and  A  /  1 2-  is  its  sine ;  hence,  OMP 

is  complementary  to  POM,  and  therefore  the  vibrations  at  M 
are  directed  along  the  line  PM.  We  conclude,  therefore,  that  the 
direction  of  the  vibrations  of  the  molecule  at  any  point  of  the  wave 
surface  is  along  the  projection  of  the  radius  vector  on  the  tangent 
plane  at  that  point. 

When  the  tangent  plane  is  normal  to  the  radius  vector,  as  is  the 
case  at  the  extremities  of  the  axes,  this  determination  is  not  appli- 
cable, but  the  direction  is  in  these  cases  easily  found.  The  plane 
OMP,  which  contains  the  radius  vector  and  the  direction  of  the 
corresponding  vibration,  is  called  the  plane  of  vibration. 

113.  Relations  between  the  Directions  of  Normal 
Propagation   of  Plane    Waves,  the    Directions   of 
Radii-  Vector es  of  the  Wave  Surface,  and  the  Direc- 
tions of  Vibrations.    By  the  preceding  theorem  we  have  seen 
that,  in  any  plane  wave  whatever,  the  normal  to  this  plane,  the 
direction  of  the  vibrations  in  this  wave,  and  the  radius  vector  drawn 
to  the  wave  surface  at  the  point  of  tangency,  are  all  contained  in 
the  same  plane.     Besides,  for  each  normal  direction  of  propagation 
of  a  plane  wave,  there  correspond  for  the  vibrations,  two  directions 
parallel  to  the  axes  of  the  elliptical  section  of  the  ellipsoid  (E) 
made  by  a  parallel  plane.     These  directions,  therefore,  being  per- 
pendicular to  each  other,  the  planes  which  contain,  at  the  same 
time,  the  same  direction  of  normal  propagation,  the  two  vibrations, 
and  the  two  corresponding  radii-vectores,  are  rectangular. 

114.  Since  the  wave  surface  has  two  nappes,  each  radius  vector 
will  give  two  directions  for  the  vibrations.     We  will  now  show  that 
the  planes  which  contain  a  radius  vector  and  the  directions  of  the 
two  corresponding  vibrations  are  also  rectangular ;  and  for  this  pur- 
pose we  shall  show  that  the  two  vibrations  which  correspond  to  the 
same  radius  ve'ctor  are  contained  in  the  two  planes  passing  through 
this  radius  vector  and  the  axes  of  the  elliptical  section,  that  a  plane 
perpendicular  to  the  radius  vector  cuts  out  of  the  ellipsoid  (W). 

115.  Let  0,  i/>,  %,  be  the  angles  made  by  one  of  the  axes  of  this 
elliptical  section  with  the  co-ordinate  axes ;  it  is  then  necessary  to 
demonstrate  that  the  three  lines  («,  ft  y),  (A,  p,  v),  (<p,  ^,  %),  are 
all  in  the  same  plane. 


78  ELEMENTS    OF    WAVE    MOTION. 

Eq.  (129),  which  gives  the  relations  existing  between  the  angles 
«,  )3,  y,  made  by  one  of  the  axes  of  ellipsoid  (E)  with  the  co-ordi- 
nate axes,  and  the  right  line  Z,  m,  n,  perpendicular  to  the  elliptical 
section,  can  be  applied  to  the  analogous  case  of  the  elliptical  section 
of  (W)  and  the  normal  radius  vector,  by  replacing  in  this  equation 

F2,  «,  ]3,  y,  I,  m,  n,  a2,  £2,  c2,  by  -^  $,  V,  X>  A>  P>  v>  ~2>  Jp>  #> 
respectively,  which  will  give 


Let  the  auxiliary  right  line  defined  by  the  angles  (A,  B,  C)  be 
drawn  perpendicular  to  the  two  right  lines  (A,  \L,  v)  and  ($,  V>,  %)  • 
we  will  have 

cos  A  cos  A  -f  cos  B  cos  p  +  cos  C  cos  v  =  0,  |          .      . 

cos  ^4  cos  <£  +  cos  B  cos  V  +  cos  C  cos  %  =  0.  j 

Replacing,  in  the  last  equation,  for  cos  </>,  cos  V>,  cos  x>  the 
quantities  proportional  to  them  in  Eq.  (158),  we  have 

a2  cos  A                 ff  cos  \i         D      c2  cos  v         n       n     /1  «nx 
-g g  cos  ^4  -\ — g ^-  cos  .6  H — g 2  cos  ^  =  0.    (160) 

Adding  this  to  the  first  of  Eqs.  (159),  we  have 
cos  A  cos  11  cos  v 


and  recollecting  that  the  relations 

cos  /I  cos  cos 


(162) 


cos  a  cos  ]3  cos  y 

exist,  we  have  finally 

cos  cc  cos  A  +  cos  ]3  cos  5  +  cos  y  cos  (7  =  0.         (163) 

Hence,  the  three  right  lines  («,  /3,  y),  (A,  /*,  v),  (0,  ^,  %),  being 
perpendicular  to  the  right  line  (A,  B,  C),  are  all  contained  in  the 


RELATING    TO    SOUND    AND    LIGHT. 


same  plane,  and  therefore  we  conclude  that  the  planes  which  con- 
tain, at  the  same  time,  the  same  radius  vector,  the  two  vibrations, 
and  the  two  corresponding  directions  of  the  normal  propagation, 
are  rectangular. 


116,   Discussion  of  the  Wave   Surface. 

Eq.  (146), 


Resuming 


—  &  (a2  +  £2)  22  +  «W  =  0, 

and  making  in  succession  x  =  0,   y  =  0,  2  =  0,  we  get   for  the 
sections  made  by  the  co-ordinate  planes  yz,  xz,  and  xy,  respectively, 

(if  +  z2  -  «2)  (%2  +  cW  -  bW)  =  0,  (164) 

=  0,  (165) 

=  0.  (166) 


Remembering  that  a  >  &  >  c,  we  see  that  the  section  in  the 
plane  yz  will  be  a  circle  whose  radius  is  «,  entirely  outside  of  an 
ellipse  whose  semi-axes  are  b  and  c.  The  section  in  the  plane  xy 
will  be  a  circle  with  radius  c,  entirely  within  the  ellipse  whose  semi- 
axes  are  a  and  b.  That  in  the  plane  xz  will  be  a  circle  with  radius 


Figure  \7, 


#,  intersecting  at  four  points  the  ellipse  whose  semi-axes  are  a  and  c. 
The  axis  of.  x  pierces  the  surface  at  distances  equal  to  ±  5  and  ±  c 
from  the  centre,  that  of  y  at  distances  of  ±  a  and  i  c,  and  that  of 
2  at  ±  &  and  ±  #. 

117.  The  surface  of  elasticity  of  two  nappes  cuts  the  axes  in  the 
same  points.  These  principal  axes  of  elasticity  have  in  turn  repre- 
sented the  square  roots  of  the  elastic  forces  developed  along  the 


80  ELEMENTS    OF    WAVE    MOTION. 

three  axes  of  elasticity,  the  principal  velocities  of  wave  propagation, 
the  axes  of  the  ellipsoids,  and  now  serve  to  fix  points  on  the  surface 
of  elasticity  and  on  the  wave  surface. 

118.  Let  Eq.  (146)  be  represented  by  L  =  0,  and  the  angles 
which  a  tangent  plane  to  the  surface  at  any  point  makes  with  the 
co-ordinate  planes  xy,  xz,  yz,  by  A,  B,  C,  respectively ;  then  we  will 
have 

1    dL  1    dL  ~       1    dL      ,.,,,-A 

cos  A  = 7-,     cos  J?  =  -• -T-,     cos  C  =  -  -  -j- ,     (167) 

o>    dz  t*   dy  u   dx  ' 

in  which,  -  =  — =.  (168) 

w  /(dirt      IdL^      IdLV 

v- W) +%)+(*) 

Taking  the  differential  coefficients  of  L  with  respect  to  x,  y,  z, 
we  will  have 


(169) 


For  y  =  0,  the  point  of  tangency  is  in  the  plane  xz,  and  we 
have 


(170) 


+  c2*2)  +  &te  (a:2  +  z*  -  a*  -  S2),   I 


the  second  equation  showing  that  the  tangent  plane  is  normal  to 
the  plane  xz. 

For  y  =  0,  the  equation  of  the  surface  gives 

a?  +  ^2  _  j2  =  o,        «2^  +  ^2  _  ^2^2  _  0;          (171) 

whence,  for  the  co-ordinates  x  and  zs  we  have 


RELATING    Tp    SOUND    AND    LIGHT. 


81 


x  — 


*  - 


(172) 


which  are  real  so  long  as  a  >  b  >  c.  There  are  then  four  points 
of  intersection  in  the  plane  xz.  Substituting  these  values  in 
Eqs.  (167),  we  obtain 


cos  A  =  -, 


cos      = 


cos      =* 


(173) 


119.  The  interpretation  of  these  indeterminate  values  of  the 
cosines  is,  that  at  the  points  considered,  a  tangent  plane  to  the 
wave  surface  may  have  any  position  whatever  with  respect  to  the 
co-ordinate  planes.     This  property  shows  that  these  points  are  the 
vertices  of  conoidal  cusps,  each  having  a  tangent  cone.     These 
points,  called  utnUlics,  belong  to  the  exterior  and  interior  nappe  of 
the  wave  surface,  just  as  the  vertex  of  a  cone  is  common  to  its 
upper  and  lower  nappes. 

120.  The  equation  of  the  right  lines  joining  these  points,  01, 
01',  through  the  centre  in  the  plane  xz  is 


z  = 

which  shows  that  the  lines 
are  normal  to  the  circular 
sections  of  the  ellipsoid 
( W).  The  lines  themselves 
are  called  axes  of  exterior 
conical  refraction. 

121.  If  tangent  lines 
be  drawn  to  the  ellipse  and 
circle,  as  MN,  M'N;,  they 
will  be  parallel  to  each 
other,  two  and  two,  and 
symmetrically  placed  with 
respect  to  the  axes  OX  and 
OZ.  The  equations  of 
these  lines  can  easily  be  shown  to  be 


(174) 


=    ± 


x±l 


Figure  18, 


-c2' 


(175) 


82  ELEMENTS    OF    WAVE    MOTION. 

and  hence  the  equation  of  the  line  drawn  from  0  perpendicular  to 
the  tangent  to  be 


'-\/rir^x'  (176) 

which  shows  that  these  lines  are  normal  to  the  circular  sections  of 
the  ellipsoid  (E). 

122.  From'  the  properties  of  this  ellipsoid,  we  see  that  a  plane 
wave  perpendicular  to  one  of  the  right  lines  MMj,  M'M/,  and  at  the 
same  time  perpendicular  to  xz,  can  be  propagated  without  altera- 
tion, whatever  may  be  the  direction  of  the  displacement  in  its  plane, 
and  that  the  velocity  of  propagation  of  this  plane  wave  is  independ- 
ent of  the  direction  of  the  displacement.     The  lines  MM,,  M'M/, 
are  called  the  optic  axes  of  the  medium,  or  axes  of  interior  conical 
refraction. 

123.  We  see,  by  comparing  Eqs.  (174)  and  (176),  that  the  lines 

01  and  OM  differ  by  the  factor  -  in  their  tangents.    This  ratio  is 

0 

always  very  nearly  unity,  and  therefore  the  lines  have  nearly  the 
same  direction. 

124.  The  planes  drawn  through  the  four  tangents  MN,  M'N', 
etc.,  perpendicular  to  the  plane  xz,  are  tangent  to  the  wave  surface 
along  the  circumferences  of  circles,  which  are  projected  in  the  lines 
MN,  M'N',  etc.     To  show  this,  let 

F(x,  y,z)  =  0  (177) 

be  the  equation  of  the  wave  surface  ;  then,  for  points  in  the  plane 
perpendicular  to  xz,  we  have 

J  -pj 

•j-  =  y  («W  +  b"f  +  <?#)  +  Vy  (#  +  y2  +  z*)  —  V  (a2  +  c2)  y  =  0. 

(178) 
which  can  be  satisfied  by  placing 

y  =  0,  (179) 

and          (a2  +  #>)  x2  -f  Wf  +  (b2  +  c2)  z*  —  #2  (a2 + c2)  =0.        (180) 

The  first  of  these  equations  gives  the  points  of  contact  in  the 
plane  xz ;  the  second  represents  an  ellipsoid.  If  we  combine  the 
equation  of  the  ellipsoid  (180)  with  the  equation  of  the  wave  sur- 
face, eliminating  #2,  the  resulting  equation  will  be  the  projection  on 
the  plane  xz  of  the  intersections  of  these  surfaces,  and  since  the  co- 


RELATING    TO    SOUND    AND    LIGHT. 


83 


ordinates  of  the  points  projected  satisfy  the  condition,  -=- -  •  =  0,  all 

y 

the  points  of  the  wave  surface  in  the  tangent  plane  which  is  per- 
pendicular to  xz,  will  be  obtained  from  this  intersection  and  projec- 
tion. The  resulting  equation  after  reduction  can  be  put  in  the 
form  of 


.  =  0.         (181) 


This  equation  can  be  satisfied  by  placing  each  factor  separately 
equal  to  zero,  and  each  will  then  be  the  equation  of  a  plane  passed 
through  one  of  the  tangent  lines  MN",  M'N',  M,N"i,  M/N/;  hence, 
each  of  the  four  planes  touch  the  surface  in  those  points  determined 
by  its  intersection  with  the  ellipsoid,  and  it  is  readily  seen  that 
these  curves  are  the  circular  sections  of  the  ellipsoid,  Eq.  (180). 
The  four  planes 


are  called  the  singular  tangent  planes  of  the  wave  surface. 

125.  The  circles  are,  in  fact,  the 
edges  of  the  conoid  al  or  umbilic  cusps, 
determined  by  the  surface  of  the  tan- 
gent cones,  reaching  their  limits  by 
becoming  planes  in  the  gradual  in- 
crease of  the  inclination  of  their  ele- 
ments, as  the  tangential  circumference 
recedes  from  the  cusp  points. 

It  thus  appears  that  the  general 
wave  surface  consists  of  two  nappes, 
the  one  wholly  within  the  other,  ex- 
cept at  four  points,  where  they  unite. 


FJEure  19. 


84  ELEMENTS    OF    WAVE    MOTION. 

Fig.  19  represents  a  model  of  the  wave  surface,  with  sections  made 
by  the  co-ordinate  planes,  so  as  to  show  the  interior  nappe. 

126.   Relations  between  the  Velocities  and  Posi- 
tions of  Plane   Waves   with  respect   to    the    Optic 

Axes*  For  each  direction  of  normal  propagation,  two  plane  waves 
travel  with  different  velocities,  determined,  as  we  have  seen,  by  the 
equation  of  the  surface  of  elasticity, 

cos2  1          cos2  m          cos2  n         . 
F2  —  a*  +  F2  —  ^  +  T2"^2  ~ 

This  equation  can  be  put  under  the  form 


cos2  1  +  («2  +  c2)  cos2  m  +  (a2+£2)  cos2  n]  F2     ,      , 
-f  W  cos2  1  +  a?c2  cos2  m  +  aW  cos2  n  =  0  ;     ^ 


and  representing  the  two  square  roots  by  F'2  and  F"2,  we  have 

F'2  +  F"2  =  (&2  +  c2)  cos2  1  +  (a2  +  c2)  cos2  m  )       ,      , 

+  («2  4-  &2)  cos2  »,  )      ^      ' 

F'2  F"2  =  W  cos2  ?  +  a2^  cos2  m  +  ^2  cos2  n.         (185) 


Let  0',  0",  be  the  angles  that  the  direction  of  normal  propagation 
makes  with  the  optic  axes,  and  0  and  180°  —  0  the  angles  that  the 
optic  axes  make  with  the  axis  x,  the  axis  of  greatest  elasticity,  then 
we  have 


cos  0  =  A  / sin  0  =  \  I (186) 

V  Cr  —  C2  V  Cr  —  C* 

cos  0'  =   cos  0  cos  I  4-  sin  0  cos  n, )      ,  „. 
cos  0"  =  —  cos  0  cos  ?  4-  sin  0  cos  w; ) 

whence,  we  get 

7   cos  0'  —  cos  0"   cos  0'  —  cos  0" 

cos  I  =  - = s — 

2  cos  0          2 

cos  0' 4  cos  0"   cos  0' 4- cos  0"  /«2  —  c2   /1QQ, 

COS  W  —  r — ; z^:  — A  /  -75 5*       (loi/j 

2  sm  0  2  V  *  —  ^ 

Substituting  these  values  of  cos  I  and  cos  ft  in  Eqs.  (184)  and 
(185),  and  replacing  cos2  m  by  1  —  cos2 1  —  cos2  ft,  we  obtain 


RELATING    TO    SOUND    AND    LIGHT.  85 

F'"  =  *      <?  -  fe 


4 

(cos  0'  -f  cos  0")2 ,  2 

=  a2  +  (?  +  (a2  —  (?)  cos  0'  cos  0", 


>   (190) 


F"F"2  =  aW  -       ~-  (cos  6?'  -  cos  0")2 


_ 

=  aW  +  ^  (cos2  0'  +  cos2  0") 


(191) 


whence, 


=  (  F'2  +  F"2)2  —  4  F'2  F"2 

=  (a2  +  612)2  +  (a2— c2)2  cos2  0'  cos2  0"— 

—  (02  _  63)2  (C0g2  (9'  +  C0g2  <9" 
—   (fl8  _  ^2)2  (!  _  Cog2  0')  (!  _ 
Q<  gi 


.   (192) 


and  finally,        F'2  —  F"2  =  (a2  —  c2)  sin  6'  sin  0".  (193) 

This  equation  establishes  the  relation  between  the  velocities  of 
the  two  plane  waves  which  belong  to  the  same  direction  of  normal 
propagation,  and  the  angles  that  this  direction  makes  with  the 
optic  axes, 

127.  The  directions  of  the  two  vibratory  motions  can  be  deter- 
mined by  means  of  the  optic  axes.  These  directions  are  parallel  to 
the  axes  of  the  elliptical  section  of  (E)  made  by  the  plane  normal 
to  the  direction  of  propagation  ;  but  the  elliptical  section  is  cut  by 
the  planes  of  the  two  circular  sections  of  the  ellipsoid  in  two  equal 
diameters  of  the  ellipse,  since  they  are  equal  to  the  radius  b  of  the 
circular  section ;  they  are  therefore  equally  inclined  to  the  axes  of 
the  ellipse.  The  optical  axes  being  normal  to  the  circular  sections, 
are  projected  on  the  plane  of  the  ellipse  in  two  diameters  whio-h  are 


86  ELEMENTS    OF    WAVE    MOTION. 

perpendicular  to  those  just  spoken  of,  and  are  therefore  also  equally 
inclined  to  the  axes  of  the  ellipse.  But  these  projections  are  the 
traces  of  the  planes  containing  the  directions  of  the  normal  propa- 
gation and  each  optic  axis.  We  therefore  conclude,  that  the  bi- 
secting planes  of  the  diedral  angle  formed  by  the  planes 
containing  the  direction  of  any  normal  propagation  and 
each  of  the  optic  axes,  are  tine  planes  of  vibration  of  the  two 
plane  waves  corresponding  to  this  normal  propagation. 

128.  The  plane  xz  being  the  plane  of  the  optic  axes,  any  direc- 
tion of  normal  propagation  in  this  plane  will  make  the  diedral 
angle  0°  and  180°,  and  hence  the  planes  of  vibration  will  be  the 
principal  plane  xz  and  a  plane  containing  y  and  the  direction  of 
propagation. 

129.  Relations  between  the  Velocities  of  Two  Kays 
which  are  Coincident  in  Direction,  and  the  Angles 
that  this  Direction  makes  with  the  Axes  of  Exterior 
Conical  Refraction.    The  expressions 


*  -  & 

and 


being  the  cosines  of  the  angles  that  the  optic  axes  make  with  the 
axes  of  x,  z,  and  making  use  of  the  analogy  existing  between  the 
ellipsoid  (E)  to  the  surface  of  elasticity,  and  the  ellipsoid  (W)  to  the 
wave  surface,  we  will  have,  by  substituting  for  a*,  tf,  c2,  in  the 

111 
above,  — 2,  ^,  -^,  the  expressions 


—  V 

and 


for  the  cosines  of  the  angles  that  the  axes  of  exterior  conical  refrac- 
tion make  with  the  axes  of  x,  z. 

If  then  r  and  r',  the  two  coincident  radii- vectores  of  the  wave 
surface,  represent  the  ray  velocities  propagated  in  the  same  direc- 
tion, and  u'  and  u"  be  the  angles  made  by  this  direction  with  the 
two  axes  of  exterior  conical  refraction,  a  discussion  in  every  way 
analogous  to  that  above  for  the  optic  axes  will  determine  the  re- 
quired relation.  This  relation  may  be  at  once  determined  by 

replacing  V,  V",  & ,  6",  in  Eq.  (193),  by  *,,  ~ ,  u',  u",  respec- 
tively ;  we  then  have 


RELATING    TO    SOUND    AND    LIGHT.  87 


-To  --  773   =    ("I)  --  9)  Sm  u' 

r2      r  2       \a2      c2/ 


130.  The  axes  of  exterior  conical  refraction  being  normal  to  the 
circular  sections  of  ellipsoid  (W),  by  a  similar  course  of  reasoning 
as  in  Art.  (127),  we  will  arrive  at  the  theorem,  that  the  bisecting 
planes  of  the  diedral  angle,  formed  by  the  planes  containing 
any  radius  vector  of  the  wave  surface  and  each  of  the  axes 
of  exterior  conical  refraction,  are  the  planes  of  vibration  of 
the  two  rays  corresponding  to  this  radius  vector. 

131.  Thus,  from  the  wave  surface  we  can  determine: 

1°.  The  position  of  the  refracted  plane  waves  by  its  tangent 
planes. 

2°.  The  direction  of  the  two  corresponding  rays  by  the  points 
of  contact  of  the  two  parallel  tangent  planes. 

3°.  The  velocities  of  the  two  rays  by  the  lengths  of  the  radii- 
vectores  drawn  to  the  points  of  contact. 

4°.  The  velocities  of  the  two  plane  waves  by  the  normals  from 
the  centre  upon  the  tangent  planes. 

5°.  The  interior  directions  of  the  molecular  vibrations  by  the 
projection  of  the  radii-vectores  on  the  tangent  planes. 

6°.  The  plane  of  vibration  by  the  plane  of  the  normals  and  vi- 
brations. 

132.  We  have  now  shown  that  when  any   arbitrary  displace- 
ment is   made   in   any  homogeneous   medium,   a  disturbance    is 
propagated  in  all  directions  from  the  origin,  and  that  it  is  materially 
affected  and  controlled  by  the  elastic  forces  developed.     In  accept- 
ing the  conclusions  which  result,  the  limitations  which  have  been 
primarily  established  must  be  kept  in  mind,  to  avoid  the  danger  of 
accepting  these  results  other  than  as  exceptional  and  governed  by 
the  admitted  hypotheses  and  by  the  accuracy  of  the  mathematical 
processes  employed.     Observation  and  experiment  are  essential  to 
ascertain  to  what  extent  the  corresponding  physical  phenomena 
conform  to  these  deductions.     They  are  to  be  used,  when  at  vari- 
ance, to  modify  the  hypotheses,  and  ultimately  through  this  modi- 
fication to  approach  nearer  and  nearer  the  true  theories  of  the 
physical  science. 

133.  The  fundamental  hypotheses  upon  which   the  foregoing 
discussion  is  in  part  based  are  as  follows  : 

1°.  The  admission  of  such  a  constitution  of  the  medium  that 


88  ELEMENTS    OF    WAVE    MOTION. 

while  it  is  variable  around  any  molecule,  it  is  similarly  variable 
around  all  the  molecules.  The  propagation  of  the  disturbance 
without  change  of  direction  of  the  vibrations,  when  the  latter  are 
excited  along  the  singular  directions  depends  on  this  assumption. 
This  inequality  of  elasticity  is  unquestionably  exhibited  in  the  phe- 
nomena of  crystallization. 

2°.  That  the  excursions  of  the  displaced  molecules  are  so  small 
that  the  resultant  elastic  forces  in  any  direction  are  proportional  to 
the  displacement.  This  implies  that  the  distances  separating  the 
adjacent  molecules  are  very  great  in  comparison  with  their  dis- 
placements. 

3°.  The  principle  of  the  coexistence  and  superposition  of  small 
motions,  by  which  any  vibration  can  be  replaced  by  others  equiva- 
lent, to  it  which  are  rectilineal. 

4°.  The  inefficacy  of  the  longitudinal  component  of  the  elastic 
force  in  light  undulations,  and  the  fact  therefore  of  transversal 
vibrations.  (The  grounds  of  this  assumption  are  to  be  given  sub- 
sequently.) 

5°.  The  correlation  of  the  total  intensity  of  the  elastic  force  to 
certain  velocities,  and  its  identity  with  that  expressed  by  the 
equation 


6°.  The  principle  of  interference,  by  which  the  motion  is  en- 
tirely destroyed  everywhere,  except  upon  certain  surfaces,  which 
may  be  regarded  as  the  loci  of  first  arrival. 

134.  The  agreement  of  the  results  obtained  by  experiment  and 
from  observation  with  the  deductions  from  the  theory  is  almost 
complete,  while  the  crucial  test  of  prediction  in  several  noted  in- 
stances leaves  but  little  doubt  of  the  truth  of  the  undulatory  theory. 
The  utility  of  the  determination  of  the  wave  surface  and  of  its 
thorough  discussion  is  thus  happily  verified,  by  its  almost  complete 
capability  of  satisfactorily  explaining  most,  if  not  all,  of  the  phe- 
nomena of  physical  optics.  While  in  the  limited  course  prescribed 
for  the  Academy  we  are  unable  to  undertake  the  complete  solution, 
we  have,  in  the  short  and  elementary  discussion  here  presented, 
obtained  sufficient  data  to  prosecute  the  study  of  sound  and  light 
to  the  extent  necessary  for  our  purposes,  and  in  this  study  we  will 
have  frequent  occasion  to  refer  to  the  foregoing  analysis. 


PART    II. 
ACOUSTICS. 


135.  The  investigations  of  physical  science  show  that  all  sensa- 
tion has  its  origin  in  the  state  of  relative  motion  of  the  molecules 
of  some  medium  with  which  the  organ  of  sensation  is  in  sensible 
contact.     Each  sensation  has  its  peculiar  organ,  which,  with  its 
nerve  system,  receives  and  transmits  molecular  kinetic  energy  to 
the  brain,  where  it  is  transformed  into  sensation.     The  motions  of 
the  molecules  are,  in  general,  vibrations,  which  are  conveyed  by 
undulations  from  the  source  of  disturbance  in  all  directions  through- 
out the  medium. 

136.  Acoustics  is  that  branch  of  physical  science  which  treats 
of  sound.     The  sensation  of  sound  usually  arises  from  the  commu- 
nication of  a  vibratory  motion  of  the  tympanic  membrane  of  the 
ear,  due  to  the  slight  and  rapid  changes  of  the  air  pressure  upon  its 
exterior  surface,  the  vibratory  motion  of  the  air  being  caused  by  the 
vibration  of  other  bodies. 

137.  The  ear  consists  essentially  of  two  parts,  one  being  in  com- 
munication with  the  external  atmosphere,  the  other  with  the  brain. 

The  first  consists  of  an  irregularly  formed  tube,  beginning  at 
the  orifice  of  the  external  ear  and  ending  at  the  pharynx.  Nearly 
midway,  the  tympanic  membrane,  or  drum-skin,  of  the  ear  crosses 
this  tube  obliquely,  separating  the  external  portion,  called  the 
meatus,  from  the  part  immediately  within,  called  the  tympanum. 
That  portion  of  the  tube  leading  from  the  tympanum  to  the  pha- 
rynx, or  cavity  behind  the  tonsils,  is  called  the  Eustacliian  tube. 
The  orifice  of  this  tube  at  the  tympanum  is  generally  closed ;  but 
the  act  of  swallowing  opens  it,  whereupon  the  air  on  both  sides  of 
the  tympanic  membrane  becomes  uniform  in  density.  These  three 
portions  of  the  first  part  of  the  ear  generally,  however,  contain  air 
differing  in  density.  In  the  meatus  the  air  responds  to  all  changes, 


90  ELEMENTS    OF    WAVE    MOTION. 

however  slight  and  rapid,  taking  place  in  the  external  atmosphere; 
while  the  air  in  the  tympanum  and  Eustachian  tube  is  not  so 
affected,  unless  communication  with  the  external  atmosphere  be 
made  as  above  described. 

138.  The  other  part,  sometimes  called  the  internal  ear,  is  sur- 
rounded by  bone,  except  in  two  places,  called  the  round  and  oval 
windows.     The  cavity  thus  formed  is  called  the   bony  labyrinth. 
The  windows  are  closed  by  membranes  which  separate  the  tympa- 
num on  the  one  side  from  the  fluid  contained  in  the  labyrinth  on 
the  other.     Connecting  the  tympanic  membrane  with  the  oval  win- 
dow is  a  series  of  small  bones,  whose  function  appears  to  be  to 
transfer  the  vibrations  from  the  former  to  the  latter.     The  laby- 
rinth is  filled  with  liquid,  having  suspended  in  it  many  membrane- 
ous bags,  also  filled  with  liquid.     Upon  the  surface  of  these  bags  are 
spread  the  terminal  fibres  of  the  auditory  nerves,  which,  by  special 
arrangements,  are  enabled  to  take  up  the  energy  communicated  to 
the  liquid  in  the  labyrinth.     The  membrane  of  the  round  window 
readily  yields  to  the  pressure  of  the  liquid,  moving  out  and  in  as 
the  oval  window  is  moved  in  and  out  by  the  transfer  of  motion 
through  the  bones  of  the  ear. 

Thus  the  energy  communicated  to  the  air  in  the  external  ear  is 
conveyed  from  the  tympanic  membrane,  through  the  series  of  small 
bones  in  the  tympanum,  to  the  membrane  of  the  oval  window, 
thence  to  the  liquid  of  the  labyrinth,  and  finally  to  the  auditory 
nerves.  How  this  energy  is  transformed  into  sensation  is  unknown. 

139.  To  represent,  graphically,  the  variations  of  air  pressure, 
we  will  make  use  of  the  curve  of  pressure,  in  which  the  abscissas 
correspond  to  the  times  and  the  ordinates  to  the  excess  of  the  pres- 
sure above  its  mean  or  average  value.     The  pressure  of  the  air,  at 
any  point,  is  assumed  to  be  measured  by  the  pressure  of  air  of  the 
same  density  and  temperature  upon  a  unit  of  area.     Then  take 


to  represent  any  curve  of  pressure  as 
o 


b 


xy 

Figure  20, 


in  which  y  =  0  represents  the  standard  or  mean   pressure,  and 


RELATING    TO    SOUND    AND    LIGHT.  91 

y  =  ±  p,  a  pressure  above  or  below  the  standard  pressure.  When- 
ever the  pressures  are  strictly  proportional  to  the  corresponding 
densities,  as  by  the  law  of  Mariotte,  the  same  curve  may  also  repre- 
sent the  curve  of  density.  If  we  now  assume  that  a  curve  similar 
to  the  above  represents  the  slight  and  rapid  changes  of  pressure  of 
the  air  in  contact  with  the  tympanic  membrane  while  the  sensation 
of  a  particular  sound  exists,  we  see  that  these  changes  do  not  in 
general  affect  the  average  pressure  of  the  air,  for  the  areas  above 
and  below  the  axis  of  the  curve  are  equal.  A  curve  is  said  to  be 
periodic  when  it  consists  of  equal  and  like  parts  continuously  re- 
peated. The  wave  length  of  a  periodic  curve  is  the  projection  upon 
the  axis  of  the  smallest  repeated  portion. 

140.  The  ear  clearly  distinguishes  between  a  musical  sound  and 
a  noise.     The  former  is  a  uniform  and  sustained  sensation,  unac- 
companied by  any  marked  alteration,  save  that  of  intensity ;  while 
the  latter  is  more  or  less  varied  and  ununiform.     When  a  sonorous 
body  is  sounding,  the  most  ordinary  examination  is  sufficient  to 
show  that  it  is  in  a  state  of  vibration.     The  vibrations  or  oscilla- 
tions of  its  parts  set  in  corresponding  motion  the  adjacent  air-parti- 
cles, which  in  turn  transmit  similar  motions  to  the  next  following 
particles,  and  so  on.     The  air,  then,  is  ever  passing  through  alter- 
nate states  of  condensation  and  rarefaction.     When  these  vibrations 
are  regular,  periodic,  and  sufficiently  rapid,  the  resulting  sound  is 
uniform  in  character  and  is  called  a  musical  tone.     If  the  resulting 
sound  arises  from  vibrations  which  are  non-periodic,  it  is  called  a 
noise.     Ordinary  observation  shows  that  few,  if  any,  noises  are  per- 
fectly unmusical ;  and  few,  if  any,  sounds  are  absolutely  unmixed 
with  noise. 

141.  Propagation   of  a    Disturbance  in  an   In- 
definite Cylinder.     Let  us  suppose  the  indefinite  cylinder  MN 
filled  with  air,  and  at  the  origin  a  piston 

p,  capable  of  rapid  to-and-fro  motion.  In 
the  first  place,  let  the  piston  be  moved  a 
distance  ds  from  p  to  p',  in  the  time  dt. 


If  the  air  were  incompressible,  it  would  be  Figure  21, 

moved  bodily  over  the  distance  ds.     But 

being  compressible,  the  air  yields  to  the  motion  of  the  piston,  and 
at  the  end  of  the  time  dt  the  compression  will  have  reached  a  posi- 


92  ELEMENTS    OF    WAVE    MOTION. 

tion  m,  so  that  the  stratum  of  air,  being  condensed  from  pm  to  p'm, 
will  exert  an  elastic  force  in  excess  of  that  due  to  its  normal  state. 
Call  this  excess  6.  The  increased  elasticity  of  p'm  will  cause  it  to 
expand  in  the  only  direction  possible,  towards  the  next  stratum 
mn,  which  in  turn  becomes  compressed.  This  second  stratum 
reacts  in  both  directions ;  on  the  side  towards  mp'  it  brings  the 
molecules  of  mp'  to  rest,  their  acquired  velocity  having  a  tendency 
to  cause  them  to  pass  beyond  their  positions  of  equilibrium :  and 
on  the  side  nr  it  compresses  the  next  stratum,  increasing  its  elas- 
ticity ultimately  by  d.  In  this  manner  the  compression  is  trans- 
mitted from  stratum  to  stratum,  throughout  the  whole  length  of 
the  cylinder. 

142.  Let  V  be  the  velocity  of  propagation  of  the  condensation, 
v  the  velocity  of  the  piston ;  then  we  have 

pp'  =  ds  =  vdt,        pm  =  ds'  =  Vdt, 
pm  —  pp'  =  p'm  =  (V  —  v)  tit. 

Supposing  Mariotte's  law  applicable,  and  P  to  represent  the 
normal  pressure,  we  have 

P  :  P  +  6  ::p'm:  pm  ::  (V—v)dt  :  Vdt', 
or  d  =  p-JL_.  (196) 

Let  p'  now  return  to  its  primitive  position  p  in  the  next  succes- 
sive dt.  The  first  layer  of  the  stratum  will  be  dilated,  occupying 
the  new  space  p'p,  and  its  pressure  P  will  become  P  —  6.  The 
elastic  force  of  the  next  layer  P  will  become,  by  its  expansion  to 

the  left,  P  —  -,  increasing  that  of  the  first  also  to  P  —  -•     But 

A  <> 

the  velocity  acquired  by  the  molecules  of  the  second  layer  will 
cause  them  to  pass  beyond  their  positions  of  equilibrium,  so  that  its 
elastic  force  will  diminish  until  it  becomes  P  —  <5,  at  the  instant 
the  elastic  force  of  the  first  layer,  continually  increasing,  becomes 
P,  its  normal  value.  The  third  layer  will,  in  turn,  act  on  the  sec- 
ond as  the  second  has  acted  on  the  first,  so  that  the  dilatation  cor- 
responding to  d  will  travel  the  distance  pm  in  the  time  dt,  during 
which  the  piston  is  retracing  its  path  p'p.  The  magnitude  of  <* 
will  evidently  depend  on  the  value  pp'  and  the  time  dt.  If  dt  be 
constant  and  6  be  varied,  the  condensations  will  vary  with  d.  The 


RELATING    TO    SOUND    AND    LIGHT.  93 

analysis  shows  that  the  compressions  and  dilatations  are  propagated 
with  equal  velocities,  and  that  these  velocities  are  independent  of 
tha  degree  of  condensation  or  of  rarefaction,  when  the  medium  is 
the  same  and  the  amplitude  is  very  small. 

143.  Let  the  prong  of  a  tuning-fork  p. . .  .p'  (Fig.  22)  be  dis- 
placed a  very  small  but  finite  distance  from  its  neutral  position  a. 
By  its  elasticity  it  will  vibrate  with  equal  displacements  on  each 
side  of  its  position  of  equilibrium.  Its  velocity  increases  from  zero  at 
p  to  a  maximum  at  a,  and  decreases  in  an  exactly  reverse  manner 
to  zero  from  a  to  p'.  Let  the  duration  of  its  motion  from  p  to  p' 
be  divided  into  equal  parts,  each  represented  by  dt,  the  epoch  cor- 
responding to  the  position  p.  From  p  the  prong  describes  unequal 
but  increasing  distances  during  the  successive  dt's  to  the  position  a, 
and  unequal  but  decreasing  distances  from  a  to  p'.  Each  corre- 
sponding compression  can  be  found  from  Eq.  (196)  by  the  substi- 
tution of  the  proper  value  of  v,  and  these  compressions  or  conden- 
sations will  be  propagated  with  a  constant  velocity  V.  While  the 
prong  is  returning  from  p'  top,  the  rarefactions  will  increase  from 
p'  to  a,  and  decrease  from  a  to  p,  and  their  values  may  be  deter- 
mined from  the  same  equation.  The  condensations  will  be  sym- 
metrically distributed  with  reference  to  the  maximum  condensation, 
neglecting  the  very  small  amplitude  vp'.  Likewise  the  rarefactions 
will  be  symmetrically  distributed  with  respect  to  the  maximum 
rarefaction. 


Figure  22. 


144.  The  positive  ordinates  of  the  curve  p'rs  represent  the  suc- 
cessive condensations,  8  being  the  position  of  the  layer  reached  by 
the  first  condensation  when  the  prong  has  arrived  at  p' ;  and  the 
negative  ordinates  pr's  will  represent  the  successive  rarefactions 
when  the  first  condensation  has  reached  the  position  u,  and  the 
prong  has  returned  to  its  primitive  position  p.  The  ordinates  of 
the  other  curves  represent  either  condensations  or  rarefactions,  as 
indicated  in  the  figure  corresponding  to  the  particular  state  and 
position  of  the  prong. 


94  ELEMENTS    OF    WAVE    MOTION. 

145.  In  the  figure,  pu,  the  length  of  the  wave  is  the  distance 
traveled  by  the  disturbance  while  the  prong  is  making  a  complete 
vibration,  and  hence  we  have,  n  being  the  vibrational  number  and 
T  the  periodic  time, 

A  =  -   =  Vr.  (197) 

n 

146.  The  mean  velocity  of  the  air  molecules  is  evidently  the 
same  as  that  of  the  vibrating  prong,  and  therefore  this  will  vary 
with  the  vibrating  body.     In  the  example  given,  the  mean  velocity 
of  the  molecules  is  2  mm.  x  256  =  0.512  m.     The  actual  velocity 
of  the  air  molecules  continually  varies,  and  at  any  time  is  propor- 
tional to  the  ordi  nates  of  the  curve  a  quarter  of  a  wave  length  in 
advance  of  the  molecules  considered.     When  the  vibrating  body  has 
simple  harmonic  motion,  the  molecular  velocity  is  given  by 

v  =  aco82n—  (198) 

147.  The  value  of  «,  the  amplitude  of  the  vibration,  diminishes 
(Art.  72)  according  to  the  law  of  the  inverse  distance  from  the  cen- 
tre of  disturbance  ;  and  for  each  value  of  a  taken  as  constant  within 
the  wave  length  we  have,  by  the  above  equation,   sensibly  exact 
values  for  the  molecular  velocity  at  any  time. 

148.  When  the  vibrations  of  the  body  are  sufficiently  frequent 
during  the  unit  of  time,  and  of  sufficient  amplitude,  the  sensation 
of  sound  arises  in  the  ear,  which,  however,  we  unconsciously  refer 
to  the  vibrating  body.     A  sonorous  wave  comprises  the  series  of 
condensations  and  rarefactions  arising  from  one  complete  vibration 
of  the  sounding  body. 

149.  The  sum  of  all  the  condensations  in  the  condensed  portion 
of  the  wave  is  represented  by  the  area  of  the  curve  p'rs,  and  if  it  be 
divided  by  the  duration  of  half  the  vibration,  the  mean  condensa- 
tion will  result.     Thus,  take  the  amplitude  of  the  oscillation  of  the 
tuning-fork,  making  128  vibrations  per  second  to  be  1  mm.,  and  the 
velocity  of  propagation  to  be  340  m.  ;  then,  from  Eq.  (196),  we 
will  have 

1 


.  P  *  _.  p 

~         340000  —1-    '      ^340000-256-       1327' 

(199) 


RELATING    TO    SOUND    AND    LIGHT.  95 

Heuce,  the  change  of  density  in  the  air,  measured  by  the 
barometric  height  due  to  the  mean  condensation,  is  not  greater  than 
that  due  to  0.0226  inches  of  mercury,  when  a  sound  corresponding 
to  128  vibrations  per  second,  and  caused  by  the  fork  under  the  sup- 
posed conditions,  is  passing. 

150.  From  the  preceding  discussion  we  see  that  we  can  neglect, 
in  general,   the  absolute  displacements  of  the  air  molecules,  and 
consider  the  change  in  pressure  and  density  as  being  alone  propa- 
gated.    Therefore,  a  file  of  elastic  balls  transmitting  motion  prac- 
tically illustrates  the  state  or  condition  of  a  series  of  air  molecules 
during  the  propagation  of  a  sonorous  wave.     An  excellent  illustra- 
tion is  also  given  by  means  of  a  chain  cord.     If  it  be  attached  at 
one  of  its  extremities  to  a  fixed  point,  and  be  held  stretched  at  the 
other,  the  successive  rings  or  spirals  will  assume  positions  of  stable 
equilibrium  with  respect  to  each  other,  determined  by  the  tension. 
These  rings,  for  the  purpose  of  illustration,  may  be  taken  to  repre- 
sent the  contiguous  air  strata  or  particles  in  an  indefinite  tube,  or 
upon  any  line  along  which  sound  is  supposed  to  be  propagated.     If 
any  ring  be  plucked,  it  will,  when  released, ^oscillate  about  its  place 
of  rest  while  the  disturbance  is  being  propagated  in  both  directions 
to  the  points  of  support.     Upon  reaching  these  points  the  dis- 
turbance will  be  divided,  a  part  proceeding  in  the  new  medium, 
and  the  remainder,  being  reflected,  will  retrace  its  path,  to  be  again 
subdivided  at  the  other  end.     This  will  continue  until  the  whole 
energy  of  the  original  disturbance  has  been  dissipated.     By  increas- 
ing the  tension  the  disturbance  will  be  more  quickly  propagated, 
and  conversely.     Now  suppose,  from  the  point  of  plucking,  lines 
be  drawn  in  all  directions,  and  the  same  phenomena  occur  on  these,, 
then  the  behavior  of  each  ring  and  the  progressive  motion  of  the 
disturbance  illustrates  what  takes  place  in  air  during  the  passage 
of  a  sound  wave  along  every  right  line  drawn  from  the  origin  of  the 
sounding  body.     In  an  isotropic  and  homogeneous  medium,  the 
disturbance  moves  with  constant  velocity,  and  the  volume  whose 
surface  bounds  the  disturbed  particles  at  any  instant  is  a  sphere 
whose  radius  is  Vt. 

151.  The  general  properties  of  any  sound  are  intensity,  pitch, 
and  quality. 

Intensity  is  that  property  by  which  we  distinguish  the  relative 
loudness  of  two  tones  of  the  same  pitch  and  quality.  We  can  also, 


96  ELEMENTS    OF    WAVE    MOTION. 

in  general,  determine  which  of  two  tones  of  different  pitch  and 
quality  has  the  greater  intensity.  The  air  particles  have  but  small 
displacements  from  their  positions  of  relative  rest,  when  the  dis- 
placement is  caused  by  the  passage  of  a  sound  wave.  The  forces 
which  urge  them  back  to  their  positions  of  rest  are  assumed  to  vary 
directly  with  the  degree  of  displacement.  In  Analytical  Mechanics, 
it  is  shown  that  the  periodic  time  of  the  air  particle  depends  only 
upon  its  mass  and  the  intensity  of  the  force  of  restitution;  and 
therefore,  in  the  same  medium,  with  given  pressure,  density,  and 
temperature,  for  the  same  exciting  cause,  the  periodic  time  will  be 
constant,  but  the  mean  velocity  of  the  air  particle  will  vary  with 
the  size  of  the  orbit.  The  kinetic  energy  in  the  moving  particle, 
varying  as  the  square  of  the  velocity,  will  therefore,  for  the  same 
exciting  cause  and  the  same  medium,  under  the  same  circumstances 
of  pressure,  density,  and  temperature,  vary  directly  as  the  square  of 
the  maximum  displacement.  By  the  law  of  the  decay  of  energy, 
the  intensity  of  the  sound  will  therefore  vary  inversely  as  the  square 
of  the  distance  from  the  origin  of  the  exciting  cause.  (Art.  72.) 

152.  Pitch  is  that  property  by  which  we  distinguish  the  posi- 
tion of  two  tones  in  the  musical  scale,  and  thereby  recognize  which 
is  the  more  acute  and  which  the  more  grave.     The  pitch  depends 
upon  the  frequency  of  the  vibration ;  the  greater  the  number  of 
vibrations  produced  by  a  sounding  body  in  a  given  time,  the  more 
acute  will  be  the  resulting  sound.     The  siren  is  an  instrument  used 
to  illustrate  this  fact.     It  consists  essentially  of  a  disk  pierced  with 
a  number  of  equidistant  holes,  through  which  air  is  forced  when  it 
is  put  in  rapid  rotation.     As  the  rotation  increases,  the  sound  grad- 
ually rises  in  pitch,  and  as  it  diminishes  the  pitch  falls  correspond- 
ingly.    If  a  coin  with  a  milled  edge,  or  a  cogged  wheel,  be  put  in 
rotation,  and  a  card  be  held  against  it,  the  same  changes  in  pitch 
will  be  observed.    In  these  cases  the  single  puff,  or  stroke  of  the 
card  against  the  coin,  or  wheel,  is  essentially  a  noise,  and  when 
these  strokes  are  multiplied  sufficiently  in  a  given  time,  the  result- 
ing effect  is  a  note  of  definite  pitch.     So  that  a  clearer  distinction 
than  that  heretofore  given  should  be  made  between  a  noise  and  a 
musical  tone.    To  this  distinction  we  will  again  refer. 

153.  The  quality  of  a  musical  tone  is  that  property  by  which 
we  can  distinguish  whether  two  sounds  of  the  same  pitch,  of  either 
equal  or  unequal  intensities,  arise  from  the  same  or  different  sono- 


RELATING    TO    SOUND    AND    LIGHT.  97 

rous  bodies.  This  property  enables  us,  within  certain  limits,  to 
distinguish  voices  and  the  various  sounds  peculiar  to  different  musi- 
cal instruments.  We  have  as  yet  only  exacted  that  a  musical  sound 
shall  be  periodic  and  regular  ;  that  is,  that  during  any  vibration  the 
successive  states  of  motion  of  the  particle  shall  recur  in  the  same 
order  as  in  each  of  the  previous  vibrations.  But  it  is  evident  that 
we  may  have  an  infinite  variety  of  periodic  motion,  and  it  will  be 
shown  that  the  quality  of  the  sound  will  varj  with  each  variation 
of  the  periodic  motion,  the  wave  length  remaining  constant. 

154.  Every  one  has  experienced  the  fact  that  more  than  one 
sound  can  be  heard  at  once.     Our  attention  can  be,  for  the  mo- 
ment, fixed  upon  any  one  of  the  many  sounds  that  are  constantly 
occurring,  and  at  the  same  time  we  may  be  conscious  of  the  exist- 
ence of  the  others.     Therefore,  the  meeting  of  sound  waves  in  the 
external  ear  does  not,  in  general,  result  in  mutual  destruction,  or 
in  essential  modification;  while,   at  the  same  time,  we  must  ac- 
knowledge that  the  air  in  contact  with  the  tympanic  membrane,  at 
any  given  instant,  can  possess  but  one  determinate  pressure  and 
density.     The  changes  in  pressure  and  density  due  to  many  exciting 
causes  must,  then,  result  from  the  superposition  and  coexistence  of 
those  arising  from  each   separate  cause,  and,  in  general,  without 
destruction  or  modification.     We  have  here  the  application  of  the 
principle  enunciated  in  Art.  204,  Mechanics. 

The  more  general  statement  of  the  law  of  the  composition  of 
displacements  would  be  that  demonstrated  in  the  principle  of  the 
parallelogram  of  forces,  but  when  the  displacements  are  infinitely 
small,  we  can  take,  rigorously,  the  resultant  displacement  to  be  the 
algebraic  sum  of  the  component  displacements.  The  diameter  of 
the  meatus  at  the  tympanic  membrane  does  not  exceed  0.25  inch, 
and 'therefore,  for  sounds  whose  sources  are  at  ordinary  distances, 
the  wave  fronts  at  the  position  of  the  tympanic  membrane  coincide 
sensibly  with  their  tangent  planes,  and  the  changes  of  density  and 
pressure  may  be  compounded  by  the  law  of  small  motions,  without 
appreciable  error. 

155.  Let  the  broken  line,  in  the  following  diagram,  represent 
the  changes  of  pressure  upon  the  tympanic  membrane  while  a  con- 
tinuous noise,  in  which  the  ear  recognizes  no  definite  pitch,  is 
sounding  for  a  small  part  of  a  second,  and  let  the  dotted  line  repre- 
sent another  noise  of  the  same  duration. 

7 


98 


ELEMENTS    OF    WAVE    MOTION. 


Figure  23, 


Then,  if  both  noises  sound  together,  the  resultant  variation  of 
pressure  will  be  represented  by  the  full  line  obtained  by  joining  the 
extremities  of  the  ordinates  found  by  taking  the  algebraic  sum  of 
the  ordinates  of  the  separate  curves. 

These  two  noises  do  not,  in  general,  unite  into  one,  but  are 
heard  distinctly  and  simultaneously,  except  in  the  case  where  the 
two  sounds  are  nearly  alike,  and  the  two  curves  nearly  similar. 
Again,  there  is  nothing  in  the  resultant  curve  to  suggest  to  the  eye 
the  nature  of  the  two  component  curves.  Hence,  the  ear  possesses 
the  property  of  separation  ;  while  the  eye,  according  to  this  method 
of  combination  and  representation,  does  not. 

156.  Let  the  component  curves  be  periodic,  two  periods  of  O'X' 
being  equal  to  three  of  0"X". 


o\ 


Figure  24, 

The  resultant  curve  OX  will  be  a  periodic  curve,  whose  repeated 
portions  are  represented  above.  An  examination  of  this  curve  by 
the  eye  gives  no  clue  to  its  components,  and  we  may  resolve  it  into 
an  indefinite  number  of  pairs  of  components,  but  one  of  which 
would  represent  the  two  notes  which  sounding  together  will  give  us 
the  resulting  effect  upon  the  ear.  But  if  the  ear  resolves  the  com- 
posite note  represented  by  OX,  it  must  resolve,  in  like  manner, 
O'X'  and  0"X".  Observation  confirms  this  deduction. 

157.  The  only  note  the  ear  is  incapable  of  resolving  is  that  of 
the  simple  musical  tone,  and  this  incapability  arises  from  the  fact 
that  such  a  tone  is  in  reality  perfectly  simple,  and  not  compound. 
The  tones  which  are  ordinarily  called  simple,  are,  in  reality,  com- 
pounded of  a  series  of  simple  tones  theoretically  unlimited  in  num- 
ber. Very  few  of  them  have  sufficient  intensity  to  be  heard ;  but 


RELATING    TO    SOUND    AND    LIGHT. 


99 


these  few  form  a  combined  note  which  is  always  the  same  under 
the  same  circumstances,  and  we  habitually  associate  them  together, 
and  perceive  them  as  a  single  note  of  a  special  character.  But  it  is 
possible,  with  certain  appliances,  to  partially  analyze  the  composite 
note  by  an  attentive  study  of  the  separate  constituents. 

Whenever  two  sounding  bodies  give  notes  whose  tones  form  con- 
sonant combinations  with  each  other,  the  difficulty  of  analysis  is 
increased;  when  the  combinations  are  dissonant,  the  analysis  is  less 
difficult. 

158.  A  noise  may  therefore  be  defined  to  be  a  combination  of 
musical  tones,  too  near  in  pitch  to  be  separately  distinguished  by 
the  unassisted  ear,  or  to  be  a  combination  of  noises,  each  of  which 
is  made  up  of  sounds  so  near  each  other  in  pitch  as  to  be  undistin- 
guishable  ;  the  separate  noises  may  be  near  or  far  apart  in  pitch. 
It  is  so  complex,  that  its  analysis  is  beyond  the  power  of  the  un- 
assisted ear.     A  simple  musical  tone,  on  the  contrary,  is  incapable 
of  resolution  by  reason  of  its  absolute  simplicity.     Hence,  strictly 
speaking,  only  simple  tones  have  pitch.     A  simple  musical  tone 
will  have  a  single  determinate  pitch.     The  pitch  of  a  musical  note 
must  then  be  taken  to  mean  the  pitch  of  the  gravest  simple  tone  in 
its  combination.     If  the  higher  simple  tones  be  successively  stopped 
out,  the  pitch,  as  defined,  will  remain  unaltered,  but  the  quality  of 
the  note  will  undergo  variations  until  the  single  musical  simple 
tone  corresponding  to  the  gravest  tone  is  reached,  beyond  which  no 
further  modification  can  take  place. 

159.  We  will  hereafter  assume,  as  the  fundamental  simple  tone, 
that  component  of  any  note  which  corresponds  to  the  regular  pe- 
riodic curve  of  the  given  pitch.     This  distinction  is  important ;  for 
it  is  evident  that  there  may  be  many  periodic  curves  of  the  same 
pitch,  and  each  may  correspond  to  musical  notes  differing  in  quality. 


Figure  25. 


100  ELEMENTS    OF    WAVE    MOTION. 

The  preceding  curves  (Fig.  25)  represent  notes  of  the  same  pitch, 
but  of  different  quality.  Helmholtz  has  shown  that  while  every 
different  quality  of  tone  requires  a  different  form  of  vibration,  the 
converse  is  not  necessarily  true;  i.  e.,  that  different  forms  of  vibra- 
tion may  not  correspond  to  the  same  quality. 

160.  We  have  seen,  page  45,  that  any  physical  condition,  such 
as  density,  pressure,  velocity,  etc.,  which  is  measurable  in  magnitude 
or  intensity,  and  which  varies  periodically  with  the  time,  may,  by 
Eq.  (195),  be  expressed  as  a  function  of  the  time.  Hence,  every 
periodic  disturbance  of  the  air,  and  particularly  such  disturbances 
as  excite  the  sensation  of  a  musical  tone,  can  be  resolved  into  its 
harmonic  vibrations. 

A  single  simple  tone  being  represented  by  the  simple  harmonic 
curve 

y'  =  a'  sin  (~  +  «'),  (200) 

\    A  I 

and  another  of  half  wave  length  by 

9.-JT/JT  \ 

(201) 


y"  =  a"sml~  +  a"\ 
\   2  / 


the  resultant  curve  will  be  represented  by 

?  +  a'\  +  a"  sin  (-™  +  «"),  (202) 


y  =  a  sn    ~ 

which  has  the  same  wave  length,  but  a  different  amplitude  and 
phase.  This  change  in  the  amplitude  and  phase  may  be  varied  at 
pleasure,  by  conceiving  the  second  curve  to  be  shifted  along  the 
axis  any  distance  from  zero  to  A,  and  again  to  pass  through  all 
values  of  the  amplitude  between  any  two  limits.  The  resultant 
curve,  in  all  cases,  will,  however,  be  a  periodic  curve  of  constant 
wave  length. 

161.  Considering  the  simple  musical  tones  which  they  represent 
then  to  be  sounded  together,  with  the  same  modifications,  it  has 
been  found  that  the  ear  can  distinguish  the  components  when  the 
attention  is  cultivated  and  directed  to  this  effect.  With  a  variation 
in  phase  only,  the  effect  on  the  ear  is  constant  and  invariable,  and 
hence  we  see  that  many  different  resultant  curves  may  represent 


RELATING    TO    SOUND    AND    LIGHT. 


101 


Phase  diff  'era  90°  at  O 


Figure  26. 

essentially  the  same  sensation.  Thus,  the  two  curves  above,  repre- 
sent the  same  compound  tone  made  up  of  the  two  simple  tones, 
although  the  forms  of  the  curves  are  quite  different.  The  resultant 
tones  are  the  same,  both  in  quality  and  in  pitch,  but  differ  in  inten- 
sity. By  combining  in  the  same  way  other  simple  tones  of  one- 
third,  one-fourth  the  wave  length,  and  so  on,  the  quality  will  be 
changed,  without  affecting  the  pitch,  as  can  be  seen  from  the 
graphical  construction,  and  heard  by  audible  experience.  In  all 
these  cases  the  untrained  ear,  by  the  aid  of  certain  appliances,  can 
always  analyze  the  resultant  sound  into  its  component  simple  tones, 
and  when  trained,  often  without  this  assistance.  When  but  one 
simple  vibration  of  sufficient  frequency  and  intensity  to  produce 
sensation  alone  exists,  no  such  analysis  takes  place. 

162.  The  investigations  of  Helmholtz  have  shown  that  the  ear 
possesses  the  property  of  analysis  of  a  single  musical  tone  into  its 
simple  musical  tones,  each  of  which  is  distinctive  in  character,  but 
which  blend  harmoniously  into  the  single  tone  when  sounded  to- 
gether. The  wave  lengths  of  these  components  are  aliquot  parts  of 
the  wave  length  of  the  fundamental,  and  the  simple  tones  are  called 
the  upper  partials  of  the  fundamental  or  prime  tone.  Hence,  from 
Art.  64  and  these  facts,  we  conclude  that,  when  several  sounding 


102  ELEMENTS    OF    WAVE    MOTION. 

bodies  simultaneously  excite  different  sounds,  the  variations  of  air 
density  and  the  resultant  displacements  and  velocities  of  the  air 
particles  in  contact  with  the  tympanic  membrane  are  each  equal  to 
the  algebraic  sum  of  the  corresponding  changes  of  density,  the  dis- 
placements and  the  velocities  which  each  system  of  waves  would 
have  separately  produced,  had  it  acted  alone. 

163.  This  analysis  by  the  ear  clearly  shows,  then,  that  the  sep- 
arate effects  of  the  simple  vibrations  are,  in  general,  neither  modi- 
fied nor  destroyed,  but  actually  exist,  and  it  remains  to  be  proved 
that  such  is  really  the  case,  independent  of  the  peculiar  sensation 
which  is  the  result  of  their  action  upon  the  ear.     Since  Fourier's 
Theorem  mathematically  demonstrates  that  any  form  of  vibration, 
no  matter  how  varied  its  shape,  can  be  expressed  as  the  sum  of  a 
series  of  simple  vibrations,  its  analysis  into  these  simple  vibrations 
is  independent  of  the  capacity  of  the  eye  to  perceive  by  examining 
its  representative  curve  whether  it  contains  the  simple  harmonic 
curves  or  not,  and  if  it  does,  what  they  are.     All  that  the  curve 
indicates  is  that  the  more  regular  its  form,  the  greater  the  effect  of 
its  deeper  or  graver  tones  in.  comparison  with  its  upper  partials. 
Before  proceeding  to  show  that  these  component  vibrations  actually 
exist  together,  and  that  each  can   affect  the  ear  or  other  sensitive 
vibrating  body,  let  us  now  establish  clearly  the  definitions  pertain- 
ing to  the  subject. 

164.  A  simple  or  pendular  vibration  is  that  which  corresponds 
to  the  complete  oscillation  of  a  simple  pendulum,  and  is  graphically 
represented  by  the  simple  harmonic  curve. 

A  simple  musical  tone  is  that  effect  produced  upon  the  ear  when 
a  sonorous  body  is  executing  simple  vibrations  only,  of  sufficient 
frequency  and  amplitude  to  be  heard.  According  to  this  definition, 
simple  tones  do  not  in  reality  exist  ;  but  in  the  vibrations  of  such 
bodies  as  tuning-forks,  the  component  vibrations  which  simulta- 
neously exist  with  that  of  the  gravest  period,  are  generally  non- 
periodic  with  it,  and  so  deficient  in  intensity  that  their  influence  is 
negligible,  and  we  may  regard  such  bodies  as  producing  simple 
vibrations  alone  without  sensible  error. 

165.  A  single  musical  tone  may  be  either  simple  or  compound. 
When  compound,  it  is  made  up  of  its  fundamental  simple  tone, 
together  with  its  upper  partial  simple  tones,  each  of  which  has  a 
frequency  of  either  twice,  three  times,  or  so  on,  that  of  its  funda- 


RELATING    TO    SOUND    AND    LIGHT. 


103 


mental.  It  is  due  to  the  vibration  of  a  single  sonorous  body  which, 
•during  its  motion,  vibrates  as  a  whole,  and  divides  also  into  parts 
which  vibrate  twice,  three  times,  and  so  on,  as  rapidly  as  the  whole. 
One  or  more  of  these  upper  partials  may  be  wanting  during  the 
vibration  ;  when  this  occurs,  the  quality  of  the  single  musical  tone 
is  correspondingly  affected. 

A  composite  musical  tone  is  composed  of  two  or  more  single 
musical  tones. 

166.  Musical  Intervals.  The  extreme  range  of  the  hu- 
man ear  lies  between  20  and  40000  simple  vibrations  per  second. 
The  corresponding  wave  lengths  are  obtained  by  dividing  the  veloc- 
ity of  sound  by  these  numbers,  and  are  approximately  54.6  feet  and 
0.0273  feet  respectively,  assuming  the  velocity  of  sound  to  be  1092 
feet  at  0°  C.  The  ordinary  sounds  heard  by  the  ear  have  a  much 
less  range ;  their  vibrational  numbers  lie  between  40  and  4000,  cor- 
responding to  wave  lengths  of  about  27.3  feet  and  0.273  feet, 
respectively.  When  a  stretched  wire  is  put  into  vibration,  and  the 
tension  continuously  undergoes  variation,  the  pitch  of  the  sound 
passes  by  continuity  from  lower  to  higher,  or  the  reverse,  and  we 
therefore  experience  the  sensation  of  a  musical  interval  between  any 
two  limiting  tones.  We  may,  then,  define  a  musical  interval  by  the 
ratio  of  the  vibrational  numbers  of  the  two  limiting  tones.  Thus, 
if  the  two  tones  correspond  to  the  vibrational  numbers  256  and 
384,  the  name  of  the  interval  is  the  fifth,  and  it  is  expressed  by  the 

o 

fraction  -•     Considering  the  simpler  ratios  that  lie  between  two 

a 

tones  whose  vibrational  numbers  are  as  1:2,  we  obtain  the  follow- 
ing musical  intervals : 


Consonant. 


Unison, 
Minor  third, 
Major  third, 
Fourth, .    . 
Fifth,     .     . 
Major  sixth, 
Octave, .    . 


Dissonant. 

Major -second,  .    .    .    9:8    '—  -§ 
Minor  second, .    .    .  10  :  9  -  =  ^ 
One-half  major  tone,  16  :  15  =  ^-f 
One-half  minor  tone,  25  :  24  =  f£ 
Comma, 81 :  80  = 


The  first  are  called  consonants,  because  the  effect  is  pleasing  to 


104  ELEMENTS    OF    WAVE    MOTION. 

the  ear  when  the  tones  of  either  of  these  intervals  are  sounded  to- 
gether. All  other  intervals  within  range  of  the  octave  are  called 
dissonants. 

167,  The  measure  of  the  musical  interval  represented  by  the 

ratio  -  is  the  log  -•     This  arises  from  the  fact  that  if  we  consider 
q  5  q 

any  three  tones  whose  vibrational  numbers  are  p,  q,  and  r,  the 
musical  interval  between  p  and  r  must  be  equal  to  the  sum  of  the 
two  intervals  between  p  and  q,  and  q  and  r.  If  the  ratios  of  the 
vibrational  numbers  were  taken  to  measure  the  intervals,  we  would 
have,  for  the  same  interval,  the  expressions 

r          ,      q   ,  r 

and      -  H 

p  p       q 

which  are  not  equal  to  each  other.     But  since 

*-  =  q~  x  T-,  (203) 

P      P      V 

we  have  log  *-  =  log  J  +  log  T-,  (204) 

and  we  may  therefore  take  the  logarithm  of  the  ratio  of  the  vibra- 
tional numbers  as  the  measure  of  the  musical  interval.  The  name 
of  any  interval,  then,  is  the  ratio  of  the  vibrational  numbers,  and 
its  measure  is  the  logarithm  of  that  ratio.  The  logarithms  are 
usually  taken  in  the  common  system. 

168.  Musical  Scales.    A  series  of  tones  at  finite  intervals 
is  called  a  musical  scale.     If  the  vibrational  numbers  are  in  the 
proportion  of  the  natural  numbers,  the  musical  scale  is  called  the 
harmonic  scale.     When  two  tones  whose  interval  is  that  of  an  octave 
are  sounded  together,  we  are  conscious  of  a  certain  sameness  of  sen- 
sation, which  is  absent  in  all  other  intervals  except  multiples  of  the 
octave.     We  may  then  assume  this  interval  as  a  natural  unit,  since 
it  gives  a  periodic  character  to  the  scale.     Whatever  properties  are 
found  with  regard  to  the  tones  in  any  octave,  occur  in  the  other 
octaves  of  a  higher  or  lower  pitch.     The  vibrational  numbers  of  the 
tones  of  the  harmonic  scale,  starting  with  a  fundamental  tone  whose 
vibrational  number  is  128,  will  be  as  follows: 

128  :  256  :  384  :  512  :  640  :  768  :  896  :  1024  :  1152  :  1280  :  etc. 


RELATING    TO    SOUND    AND    LIGHT.  105- 

169.  Examining  these  numbers,  we  see  that  each  interval  in 
any  octave  is  divided,  in  the  succeeding  octave,  into  two  intervals 
which  can  be  obtained  from  the  equation 

7M;JL  _  2n  +  1       2^  +  2 
n  2n       *  2^M' 

n  being  the  natural  number  which  marks  the  position  of  the  first 
tone  of  the  lower  interval  in  the  harmonic  scale.  Thus  we  see  that 
the  interval  128  :  256,  or  the  octave,  is  divided  in  the  next  octave 

.   ,        ,  2n  +  1        3        384       ,  2n  +  2 

into  two  intervals  represented  by  — =  -  =  -  -  and  - 

2n  2        256  2n  +  I 

=  -  =  --— .     The  first  interval,  256  :  384,  in  the  second  octave  is 

9      O-   1 

divided  into  the  two  intervals  corresponding  to  -         -  =  -  and 

9  9  K 

-  =  •=  in  the  third  octave ;  the  second  interval,  384  :  512,  in 
An  -f-  1        o 

the  same  octave,  is  in  like  manner  divided  into  — -  ^—  =  -  and 
2n  ,  o  8  2ra  6 

-  =  ^  in  the  third.     The  first  interval,  512  :  640,  in  the  third 

+     .  9  10 

octave,  is  subdivided  in  the  fourth  octave  into  ^  and  — ,  and  so 

o          y 

on.  Arranging  all  the  intervals,  with  their  corresponding  subdi- 
visions in  the  next  higher  octave,  we  have 

2 
1st  octave,  128  :  256,  interval  -,  subdivided  in  2d  octave  into 

256  :  364  =  |    and     384  :  512  =  £ ; 
/c  o 

o 

2d  octave,  256  :  384,  interval  -,  subdivided  in  3d  octave  into 

/c 

512  :  640  =  |    and     640  :  768  =  | ; 

4. 
2d  octave,  384  :  512,  interval  ^,  subdivided  in  3d  octave  into 

o 

768  :  896  =        and    896  :  1024  =     ; 


106  ELEMENTS    OF    WAVE    MOTION. 

3d  octave,  512  :  640,  interval  j,  subdivided  in  4th  octave  into 

n  10 

1024  :  1152  =  5    and     1152  :  1280  =  ^  ; 


n 

3d  octave,  640  :  768,  interval  -,  subdivided  in  4th  octave  into 

11  12 

1280  :  1408  =  ^    and    1408  :  1536  =  — 

Thus  every  interval  in  the  harmonic  scale  is  divisible  into  two 
other  intervals,  whose  ratios  are  those  of  consecutive  numbers  in  the 
next  higher  octave. 

170.  Perfect  Accords.    A  perfect  accord  is  a  series  of  three 
tones,  called  a  chord,  which,  sounded  simultaneously,  give  a  partic- 
ularly pleasing   sensation  to  the  ear.     The  perfect  major  accord 
•consists  of  the  three  tones  called  the  tonic,  the  middle,  and  the  dom- 

K  q 

mant,  whose  intervals  are  a  major  third  and  a  fifth,  or  -  and  -• 

n 
The  perfect  minor  accord  is  composed  of  a  minor  third,  -  ,  and 

a  fifth,  |. 

A 

171.  The  Diatonic  Scale.    The  tones  of  this  scale  are 
usually  designated  by  letters  or  symbols,  as  follows  : 

C:D:E:F:G:A:B:£:d:  etc. 

ut  or  do  :  re  :  mi  :  fa  :  sol  :  la  :  si  :  do  :  re  :  etc. 
Forming  the  perfect  major  accord  on  C  as  a  tonic,  we  will  have 
C    :    E    :    G, 


4         2 

Forming  similar  chords  with  C  and  G,  by  making  0  a  dominant 
and  G  a  tonic,  we  will  have 


RELATING    TO    SOUND    AND    LIGHT.  107 

F!  :    A,   :    C,  G    :      B     :    d, 

25  3          15          9 

36  2         ~S          4* 

Arranging  these  three  chords  in  order  of  their  pitch,  we  find 
Fj  :    A!  :    C    :    E    :    G    :     B     :     d, 

25  5         3         15         9 

36  4         2         T         V 

which  is  a  musical  scale  of  seven  notes,  rising  one  above  another  by 
alternate  major  and  minor  thirds. 

Replacing  in  this  scale  Fx,  At,  by  their  higher  octaves,  and  d 
by  its  lower  octave,  which  is  permissible,  and  arranging  in  order, 
we  have 

C:D:E:F:O:A:     B     :     c, 

9543515 
84323          8 

w^hidi  is  known  as  the  diatonic  scale.  The  names  of  the  intervals 
heretofore  used  are  now  seen  to  come  from  the  position  of  the  notes 

g 

in  this  scale  with  reference  to  the  tonic ;  thus,  the  interval  ^  is  a 

o 

major  second,  the  interval  -  a  major  third,  ^  a  fourth,  -  a  fifth, 

and  so  on.  The  first  tone  in  the  scale  is  called  the  tonic,  the  fifth 
the  dominant,  and  the  fourth  the  subdominant.  Taking  the  vibra- 
tional  number  of  the  tonic  C  to  be  24,  we  have  the  corresponding 
vibrational  numbers  of  the  diatonic  scale, 

C   :   D  :   E  :   F   :   G  :   A  :   B  :    c, 

24  :  27  :  30  :  32  :  36  :  40  :  45  :  48. 

172.  The  vibrational  numbers  of  the  other  octaves  are  obtained 
from  these  by  constantly  doubling  or  halving  them,  according  as 
we  ascend  or  descend,  the  letters  being  properly  accented  to  indi- 
cate in  which  octave  the  series  is  taken.  Theoretically,  the  tones 
of  the  diatonic  scale  above  belong  to  the  harmonic  scale,  whose  fun- 
damental tone  has  one  vibration  per  second.  This  fundamental 


108  ELEMENTS    OF    WAVE    MOTION. 

32        /2\5 
tone  is  five  octaves  below  the  subdominant ;  for  —  =  (    )  •    We 

will  hereafter  take  the  octave  whose  tonic  corresponds  to  256  vibra- 
tions for  that  of  comparison,  because  Scheibler's  tonometer,  which 
we  use  in  illustration  in  the  lectures  on  this  subject,  is  based  on 
that  tonic. 

173.  The  relation  of  the  successive  tones  of  the  harmonic  scale 
to  any  tone  assumed  as  a  fundamental  is  as  follows  ;  taking  as  the 
prime  that  whose  vibrational  number  is  256,  we  have 

Prime  or  fundamental,  256  vibrations,  or  c ; 
1°  Harmonic,  512        "  "  cr,  octave; 

2°         "  768         "  "  g',  fifth  in  1st  octave ; 

3°         "  1024         "  "  c",  second  octave ; 

4°         "  1280         "  "  e",  maj.  third  in  2d  oct.; 

5°         "  1536         "  "  g",  fifth  of  2d  octave; 

6°         "  1792         "  "  a"  +  ,  lying  between  6th 

and  7th  of  2d  oct. ; 
7°         "  2048         "  "  c'",  third  octave; 

and  so  on.  These  harmonics  are  called  overtones  or  upper  partials, 
and,  as  seen  above,  bear  a  close  relationship  to  the  prime.  When 
the  prime  is  sounded  and  the  upper  partials  exist  at  the  same  time, 
the  resulting  tone  will  have  a  determinate  quality.  And  if  the  par- 
tials be  successively  stopped  out,  the  quality  will  undergo  a  change, 
until  we  reach  the  simple  tone  due  to  the  prime  alone.  The  suc- 
cessive curves  which  represent  these  tones  graphically  will  approx- 
imate gradually  to  that  of  the  harmonic  curve  of  the  wave  length 
of  the  prime,  which  it  ultimately  reaches  when  all  of  the  partials 
are  wanting.  The  wave  lengths  of  the  above  curves  are  each  equal 
to  that  of  the  prime. 

174.  It  can  be  experimentally  shown  that  a  stretched  cord, 
when  plucked  from  its  position  of  rest,  will  give  a  compound  tone, 
which  is  made  up  of  its  fundamental  united  to  some  of  its  overtones. 
The  educated  ear  can  readily  distinguish  the  existence  of  these 
simple  tones,  which,  sounding  together,  determine  the  quality  of 
the  compound  tone.     But  to  demonstrate  to  the  untrained  ear  the 
existence  of  these  partial  tones,  it  is  necessary  to  make  use  of  cer- 
tain  appliances   called  resonators,   whose  action   depends  on  the 


RELATING    TO    SOUND    AND    LIGHT.  109 

principle  of  sympathetic  resonance.  These  consist  of  metal  or  other 
hollow  bodies,  generally  spherical  in  form,  closed  except  at  two 
places;  one  of  the  openings  is  to  permit  the  mass  of  air  within  to 
be  affected  by  the  vibration  of  the  air  without,  and  the  other  to  per- 
mit the  air  within  to  be  brought  into  near  contact  with  that  in  the 
aperture  of  the  ear. 

175.  Sympathetic   Resonance.      If  a  body  capable  of 
taking  up  an  oscillatory  motion  of  definite  period  be  subjected  to  a 
series  of  periodic  impulses,  whose  period  is  the  same  as  that  of  the 
body  considered,  the  aggregate  effect  will  in  time  become  sensible, 
however  weak  the  impulses  may  be.     But  if  the  period  of  the  im- 
pulses be  even  slightly  different  from  that  of  the  body,  the  resultant 
effect  will,  in  general,  never  become  appreciable;  for,  while  the 
kinetic  energy  is  increased  by  the  elementary  quantities  of  work  due 
to  the  impulses  applied,  soon  the  succeeding  impulses  will  be  deliv- 
ered in  a  direction  contrary  to  the   motion  of  the  body,  and  the 
kinetic  energy  will  be  correspondingly  diminished.     The  maximum 
energy  can  then  never  exceed  a  small   definite  quantity,  and  in 
reaching  this  state  the  body  will  pass  through  alternations  of  rest 
and  motion.     To  determine  the  effect  of  any  periodic  impulse  upon 
a  body  capable  of  being  put  into  vibration,  we  have  the  following 
rule,  due  to  Helmholtz:  Resolve  the  periodic  motion  of  the  impulse 
into  its  component  simple  pendular  vibrations ;  if  the  periodic  time 
of  any  one  of  these  vibrations  is  equal  to  the  periodic  time  of  the 
body  acted  upon,  sensible  vibration  will  result,  and  not  otherwise. 

176.  Now  consider  the  mass  of  air  within  the  tube  AB,  while  a 
simple  vibratory  motion,  due  to  a  sim- 
ple  tone,   occurs  in   the  external  air.  ~  \   / 

Let  V  be  the  velocity  of  wave  propaga-       

tion  in  the  air  under  consideration,  and 

n  the  vibrational  number  of  the  body. 

Then,  during  the  first  semi-vibration,  Figure  27. 

the  molecules  at  B  describe  half  their 

orbits  while  undergoing  condensation,  which  is  transmitted  through 

the  intervening  molecules  to  A  and  back  to  B,  provided 


110  ELEMENTS    OF    WAVE    MOTION. 

During  the  second  semi-vibration,  the  rarefaction  at  B  will  be  trans- 
mitted in  the  same  manner,  and  the  orbits  at  B  will  be  completed. 

V 

Should  BA  be  either  >  or  <  — ,  the  second  impulse  would  reach 

MI 

B  after  or  before  its  molecular  orbits  had  been  completed.  Under 
these  circumstances,  succeeding  impulses  would  in  a  short  time 
reduce  the  displacements  of  the  molecules  to  zero,  and  never  permit 
them  to  attain  an  appreciable  value,  and  therefore  the  vibration  of 
the  air  column  would  not  give  a  sound  of  appreciable  intensity. 
But  if,  on  the  contrary,  the  impulses  were  of  the  same  periodicity 
as  the  air  molecules,  each  successive  impulse  would  add  to  the  first 
displacement,  and  this  addition  would  continue  until  the  work  of 
the  resistances  developed  was  exactly  equal  to  the  increment  of 
energy  caused  by  each  impulse.  The  displacements  of  the  mole- 
cules would  then  have  attained  their  maximum  value,  and  the 
resulting  sound  a  fixed  intensity. 

177.  Each  confined  mass  of  air  has  a  particular  periodicity,  and 
each  of  the  resonators  of  Helmholtz  is  carefully  contrived  to  respond 
to  a  given  periodicity  of  vibratory  motion.     If,  then,  by  the  rule 
above  given,  any  composite  sound  exist,  and  one  of  these  resonators 
be  applied  to  the  ear,  the  resonant  effect  will  indicate  whether  the 
simple  tone  corresponding  to  the  resonator  is  present  or  absent  in 
the  composite  sound.     This  and  analogous  experiments  show  that 
sympathetic  vibration  is  not  due  to  any  property  peculiar  to  the  ear, 
but  that  it  is  a  mechanical  effect  separate  and  distinct  from  the 
sense  of  audition. 

178.  The  energy  of  motion  depending  upon  the  mass  and  ve- 
locity, we  see  clearly  that  of  two  sounding  bodies,  vibrating  with 
the  same  amplitude,  the  smaller  mass  will  more  quickly  give  up  its 
energy  to  the  surrounding  air  and  sooner  cease  sounding.     Tuning- 
forks  being  generally  made  of  steel,  will,  when  put  into  rather 
strong  vibration,  continue  sounding  for  a  reasonable  length  of  time. 
When  mounted  upon  their  resonant  boxes,  the  latter  containing  a 
mass  of  air  capable  of  vibrating  in  unison  with  it,  they  affect  larger 
masses  of  air  than  when  not  so  mounted,  and  come  more  quickly  to 
rest;  but  the  sound  will  have  greater  intensity,  and  can  the  more 
readily  be  used  to  study  the  phenomena  of  sympathetic  resonance. 
If  such  a  tuning-fork  be  in  the  vicinity  of  a  vibrating  sounding 
body  whose  sound  contains  the  tone  of  the  fork,  the  latter  will  in 


RELATING    TO    SOUND    AND    LIGHT.  Ill 

time  indicate  the  fact  by  coming  into  sympathetic  vibration.  The 
analysis,  then,  of  any  composite  note  can  be  practically  made  by 
means  of  a  sufficient  number  of  such  forks,  whose  vibrationul  num- 
bers embrace  all  the  simple  notes  of  the  composite  sound.  Con- 
versely, the  synthesis  of  a  composite  note  can  be  effected  by  setting 
in  vibration  all  the  forks,  with  proper  amplitudes,  which  the  analy- 
sis indicates  belong  to  the  note  in  question. 

179.  When  plates,  bells,  strings,  etc.,  are  put  into  vibration, 
they  may  either  vibrate  as  a  whole,  or  separate  into  parts  which 
vibrate  two,  three,  four,  or  more  times  as  rapidly ;  or  both  of  these 
conditions  may  occur  simultaneously.     Each  of  the  simple  periodic 
vibrations  has  an  actual  existence,  and  corresponds   to   a  single 
musical  tone  of  definite  pitch,  which  may  be  recognized  as  above 
described. 

180.  In  listening  for  any  simple  tone  in  the  composite  note,  it 
is  important  to  clearly  fix  the  attention  upon  the  special  tone  whose 
existence  is  to  be  determined,  and  for  this  purpose  the  tone  should 
be  sounded  alone  before  listening  for  it  in  the  composite  note. 
When  sufficiently  practiced  in  this  manner,  the  ear  can  readily 
acquire  the  faculty  of  detecting  them  without  the  use  of  resonators. 

181.  By  means  of  the  monochord,  which  consists  essentially  of 
a  string  stretched  over  two  bridges  on  a  sounding-box,  we  can  verify 
the  simultaneous  existence  of  the  prime  and  upper  partials,  and 
estimate  the  influence  of  the  latter  in  affecting  the  quality  of  the 
sound.     The  theory  of  vibrating  strings  shows  that  the  frequency 
of  vibration  of  the  same  string  under  the  same  tension  is  inversely 
proportional  to  its  length.     Plucking  the  string  at  its  centre,  the 
resulting  tone  will  be  that  of  its  prime,  modified  by  some  of  the 
upper  partials,  those  of  the  latter  being  absent  that  require  the 
middle  point  as  a  point  of  rest.     By  a  movable  bridge,  the  string 
can  be  divided  into  its  aliquot  parts,  which  being  set  in  vibration, 
will  give  the  upper  partials  in  succession.     Becoming  thus  acquaint- 
ed with  these  simple  tones,  we  can  verify  their  presence  or  absence 
in  each  special  case.     For  example,  if  the  string  be  plucked  at  one- 
fourth  its  length,  theory  requires  the  presence  of  the  first  upper 
partial  with  the  prime,  and  the  fact  will  be  made  manifest  by  damp- 
ing the  string  at  the  middle  point  immediately  after  plucking,  when 
the  octave  will  sing  out,  no  longer  encompassed  by  the  prime. 


112  ELEMENTS    OF    WAVE    MOTION. 

182.  These  and  the  facts  of  sympathetic  resonance  show  that 
the  analysis  of  all  resonant  motion  into  simple  pendular  vibrations 
is  real  and  actual,  and  that  any  other  analysis  is  highly  improbable. 
The  analogous  property  of  the  ear  is  expressed  by  the  law  of  G.  S. 
Ohm,  viz.,  that  the  human  ear  perceives  pendular  vibrations 
^alone-  as  simple  tones,  and  resolves  all  other  periodic  motions 
of  the  air  into  a  series  of  pendular  vibrations,  hearing  the 
simple  tones  which  correspond  to   these  simple  vibrations. 
We  may  therefore  conclude,  that  in  all  cases  whenever  any  motion 
of  the  air  caused  by  a  sounding  body  contains  a  simple  vibration  of 
.the  same  periodicity  as  that  of  any  other  body,  the  latter  will  in 
time  take  np  a  vibratory  motion  which,  if  of  sufficient  intensity, 
will  affect  the  ear  with  a  simple  musical  tone  of  a  definite  pitch  ; 
.and  the  mechanical  effect  of  vibration  will  ensue,  whether  it  be  of 
sufficient  amplitude  to  produce  a  sonorous  effect  or  not. 

183.  Velocity  of  Sound  in  any  Isotropic  Medium. 

The  air  is  the  medium  of  transfer  to  the  ear  of  the  vibratory  mo- 
tion of  a  sounding  body.  Under  a  given  temperature  and  density, 
its  elastic  force  is  constant  in  all  directions,  and  it  is  therefore  an 
isotropic  medium.  Being  compressible,  the  motions  of  its  mole- 
cules, during  the  passage  of  a  sound  wave,  are  to  and  fro  along  the 
line  of  wave  propagation.  They  are  then  longitudinal  vibrations, 
and  Eq.  (116),  for  the  velocity  of  wave  propagation,  for  waves  with 
.such  vibrations  in  an  isotropic  medium,  is 


sin2 


184,  The  wave  lengths  of  sound  in  air  can  never  be  greater 
than  54.6  ft.,  nor  less  than  0.027  ft.  ;  for  the  usual  sounds  the  limits 
are  27.3  ft.  and  0.273  ft.,  at  0°  0.  In  the  above  equation,  Az  is  the 
distance  separating  two  adjacent  molecules,  and  without  knowing 
its  absolute  value  for  any  degree  of  pressure,  we  may  say  that  A, 
even  in  the  minimum  sound  wave,  is  very  great  with  respect  to  A#. 
Therefore  the  arc  is  approximately  zero,  and  may  be  substituted  for 

sin--—  -  without  appreciable  error.    We  then  have 


RELATING    TO    SOUND    AND    LIGHT.  113 


(207) 

=  !*•[*« 

Hence,  with  the  supposition  of  small  displacements,  etc.,  the 
velocity  of  wave  propagation  of  sound  in  air  is  theoretically  inde- 
pendent of  the  wave  length,  and  all  sounds,  whether  grave  or  acute, 
will  travel,  in  air  of  constant  pressure  and  temperature,  with  equal 
velocities. 

Omitting  the  term  containing  A.T4  as  being  small  compared  with 
that  of  which  Az2  is  a  factor,  and  replacing  0  (r)  by  its  equal 

^  ^     ,  we  have  -i         fi  \ 

r  F2  =  i  I,p  £12  AZ*.  (208) 

/v  /* 

185.  Let  E  represent  the  modulus  of  longitudinal  elasticity  of 
air,  P  the  barometric  pressure,  /  the  length  of  the  air  column  with- 
out pressure,  and  A  the  compression  due  to  P.     Then,  by  Eq.  (I), 
we  have 

E={P.  (209) 

Since,  if  the  pressure  P  be  removed,  the  expansion  would  be 
indefinitely  great,  the  compression  A  is  sensibly  equal  to  I,  and 
therefore 

E  =  P  ;  (210) 

that  is,  the  elastic  force  of  the  air  is  that  due  to  the  barometric 
pressure  on  the  unit  area. 

186.  In  Eq.  (208),  i  ^f(r)  is  the  acceleration  due   to   the 

a 

aggregate  elastic  forces  developed  in  the  molecules  \i  by  the  arbi- 
trary displacement  of  the  molecule  m,  and  reciprocally  is  the  elastic 
acceleration  of  m ;  hence  we  have,  by  multiplying  by  m,  the  inten- 
sity of  the  elastic  force  acting  on  m, 


8 


114  ELEMENTS    OF    WAVE    MOTION. 

Aic 
-  is  the  cosine  of  the  angle  made  by  this  force  with  the  axis  of  x, 

which,  since  the  medium  is  isotropic,  is  equal  to  unity;  multiply- 

ing the  elastic  intensity  on  m  by  the  factor  —  ^,  we  have  the  elastic 
intensity  on  the  unit  area,  or 


whence,  Zp/(r)  =  -  (212) 


Substituting  in  Eq.  (208),  we  have 

V  =  ***;  (213) 

m 

Arz8  is  the  volume  of  the  molecule,  and  replacing  it  by  its  equal 
yr,  and  extracting  the  square  root,  we  will  have,  finally,  for  the 

velocity  of  wave  propagation  in  air  or  any  gas,  subjected  to  the  law 
of  Mariotte, 


4 

or  directly  proportional  to  the  square  root  of  the  ratio  of  the  elas- 
ticity of  the  medium  to  its  density. 

187,  This  conclusion  is  deduced  on  the  hypothesis  of  the  direct 
ratio  of  the  elastic  force  to  the  density,  and  if  the  law  of  Mariotte 
were  true  for  all  circumstances  of  pressure,  temperature,  and  den- 
sity, this  theoretical  velocity  and  the  actual  velocity  determined  by 
experiment  would  perfectly  accord.     But  this  relation  is  true  only 
for  a  perfect  gas  and  for  constant  temperature. 

188.  The  relation  of  these  two  parameters  of  air,  considered  as 
a  perfect  gas,  are  given  by  the  following  formulae  : 


p'd=pd', 

p'd  =  pd'  (I  +  «0), 

<217> 

in  which  p  and  p'  are  respectively  the  old  and  the  new  pressures  or 


RELATING    TO    SOUND    AND    LIGHT.  115 

the  corresponding  elastic  forces,  d  and  d'  the  old  and  the  new  den- 
sities; «  the  coefficient  of  expansion,  a  constant,  and  equal  to  ^-y 
for  Centigrade  scale ;  and  0,  degrees  of  temperature  Centigrade. 

The  first  of  these  equations  is  the  mathematical  expression  of 
Mariotte's  law ;  the  second,  of  that  of  Charles  or  Gay-Lussac ;  and 
the  third,  of  that  of  Poisson.  The  gas  to  which  these  equations  are 
applicable  is  supposed  to  be  a  perfect  fluid,  devoid  of  friction,  and 
to  have  the  pressure  at  each  point  uniform  in  all  directions. 

The  temperature  is  supposed  constant  during  all  changes  of 
pressure  and  density  in  Mariotte's  formula,  while  in  that  of  Charles 
the  gas  takes  the  pressure  and  density  determined  by  the  change  of 
temperature.  The  formula  of  Poisson  supposes  the  gas  subjected 
to  sudden  changes  of  density,  and  that  the  heat  developed,  whether 
considered  positively  or  negatively,  is  not  conveyed  by  radiation  or 
conduction  to  other  bodies,  or,  in  other  words,  that  the  quantity  of 
heat  in  the  gas  is  constant.  Remembering  that  sudden  condensa- 
tion in  air  or  gas  produces  heat,  and  sudden  rarefaction  cold,  and 
assuming  that  these  alternations  are  so  rapid  that  neither  the  heat 
nor  the  cold  is  conveyed  to  the  other  particles,  within  the  volume 
considered,  much  beyond  the  point  at  which  they  originate,  we  see 
that  this  heat  and  cold  will  produce  an  elastic  force  of  greater  in- 
tensity than  that  in  either  of  the  other  two  cases;  therefore  the 
value  of  the  velocity  of  propagation  will  be  greater  than  that  given 
in  Eq.  (214),  which  was  deduced  under  the  supposition  of  the  sim- 
ple ratio  of  the  elastic  force  to  the  density  expressed  by  ^-  It 

CL 

might  be  supposed  that  the  influence  of  the  heat  produced  in  the 
condensation  of  the  sound  wave  would  be  neutralized  by  the  cold 
produced  in  the  rarefaction,  and  that  therefore  the  resultant  effect 
would  be  zero.  This,  however,  is  not  the  case;  for  the  heat  in  the 
condensation  has  increased  the  difference  of  elastic  force  between 
the  condensed  stratum  and  the  one  in  its  front,  and  hence  has  in- 
creased the  velocity,  while  the  cold  in  the  rarefaction  has  caused  an 
equal  difference  between  the  rarefied  stratum  and  the  one  in  rear, 
and  has  thus  added  an  equal  increment  of  velocity  to  this  portion 
of  the  wave.  This  is  true  for  each  stratum  affected  by  the  sound 
wave.  Hence  the  disturbance  passes  each  stratum  of  the  condensed 
and  rarefied  portions  with  the  same  velocity,  and  this  may  be  re- 
garded as  the  velocity  of  the  wave. 


116  ELEMENTS    OF    WAVE    MOTION. 

189.  Since  the  vibrational  number  of  sound  waves  varies  be- 
tween 20  and  40000,  for  the  extreme  limits,  the  alternate  condensa- 
tions and  rarefactions  occur  with  sufficient  rapidity  to  necessitate 
the  application  of  the  formula  of  Poisson  for  the  determination  of 
the  velocity  of  sound  in  air  and  in  other  gaseous  media. 

190.  Pressure  of  a  Standard  Atmosphere.     Let  p 

be  the  pressure  of  the  atmosphere  when  the  barometric  column  cor- 
responds to  76cm.,  the  mercurial  density  being  13.5962,  and  g 
981  dynes  ;  we  then  have 

p  =  981  x  13.5962  x  76  =  1.01368  x  106  dynes,         (218) 

as  the  corresponding  pressure  of  the  atmosphere  upon  a  square  cen- 
timetre. But  since  the  density  of  mercury,  referred  to  the  stand- 
ard at  the  same  locality,  is  independent  of  the  locality,  and  hence 
independent  of  g,  we  may  assume  as  the  standard  atmosphere  that 
whose  pressure  on  the  square  centimetre  at  all  localities  is  equal  to 
106  dynes.  Hence, 

p  =  g  x  dm  x  li  =  106  dynes.  (219) 

By  substituting  in  this  equation  the  value  of  g  for  the  latitude 
of  the  place,  and  solving  with  reference  to  h,  we  will  determine  the 
barometric  height  corresponding  to  the  standard  atmosphere  at  that 
locality;  g  varies  from  978.1  dynes  at  the  equator  to  983.11  dynes 
at  the  pole. 

191.  Height   of  the    Homogeneous    Atmosphere. 

If  the  atmosphere  be  supposed  replaced  by  an  atmosphere  of  uni- 
form density  D,  as  that  of  standard  dry  air  at  0°  C.,  and  height  IT, 
exerting  the  same  pressure,  H  may  be  obtained  from  the  equation 

p  =  g-D-II  —  106  dynes;  (220) 

from  which  we  have 


=  7989.40  m.  =  26212.18  ft., 
which  is  constant  at  the  same  locality,  for  the  same  temperature 


RELATING    TO    SOUND    AND    LIGHT.  117 

and  barometric  height.     If  the  temperature  become  6°  C.,  we  have, 
by  the  law  of  Charles  or  Gay-Lussac, 

H'  =  (1  +  «0)  H  =  Ha  T,  (222) 

in  which  r  is  the  absolute  temperature,  and  a  the  coefficient  of 
expansion. 

192.   Eeplacing  the  elastic  force  E  by  its  equal,  in  terms  of  the 
homogeneous  atmosphere,  in  Eq.  (214),  we  have 


which  is  Newton's  formula  for  the  velocity  of  sound  in  air.  Mak- 
ing r  =  273°,  corresponding  to  zero  Centigrade,  and  g  =  981  dynes, 
we  have 


V  =  V  7.9894  x  105  x  981  =  2.8  x  104  =  280.0  metres.     (224) 
For  any  other  temperature,  we  have 


V  =  V7.9894  x  105  x  981  x  «T  =  280  Vl  +  «0.        (225) 

193.  These  values  of  the  velocity  of  sound  in  air  are  about  one- 
sixth  less  than  those  determined  by  experiment,  the  discrepancy 
being  due  to  the  supposition  that  Mariotte's  law  expresses  the  rela- 
tion of  pressure  and  density.  The  law  of  Poisson  is,  however,  appli- 
cable ;  hence  we  have 

(D'\y 

P  =P 


differentiating,  dp'  =  yp  —    , 


Whence  we  see  that  when  a  sound  wave  is  passing  through  air, 
the  ratio  of  the  increment  of  the  elastic  force  to  that  of  the  density 
is  equal  to  the  ratio  of  the  elastic  force  to  the  density,  multiplied  by 
the  constant  y.  The  value  of  y  can  be  determined  from  a  direct 
observation,  by  accurately  measuring  F,  a,  and  6,  and  substituting 
in  the  equation 


118  ELEMENTS    OF    WAVE    MOTION. 

io4  x  2. 


=  y  r  ^7  = 


and  solving  with  respect  to  y.  Its  value  has  been  found  to  be, 
approximately,  1.41  for  all  simple  gases  not  near  their  points  of 
liquefaction.  The  final  formula,  therefore,  is 

y  F=  338.64  m.xVrq^  ) 

=  1091.35  ft.  X  A/1  +  .003660°,  ) 

for  the  velocity  of  sound  in  air  at  the  locality  where  g  =  981  dynes, 
barometric  height  76  cm.,  and  temperature  6°  Centigrade. 

194.  At  West  Point,  assuming  the  barometric  height  to  be 
76  cm.,  and  g  =  980.3  dynes,  we  have,  for  the  velocity  of  sound 
in  air  at  any  temperature, 


V  =  v'980.3  x  7.9894  x  1.41  x  IO5  x  «T 

=  332.3  m.  x  VF+~^  •         (229) 

=  1090.23  ft.  x  Vl  +  «0. 

Since  the  value  of  a  =  ^T,  we  see  that  the  velocity  increases 
nearly  2  feet  for  each  degree  Centigrade,  and  hence  is  greater  in 
warm  than  in  cold  weather,  all  other  things  being  equal.  At 
60°  F.,  we  may  take  the  velocity  of  sound  in  air  to  be  approxi- 
mately 1123  feet  per  second. 

195.  The  value  of  the  velocity  of  sound  in  any  gas  can,  in  like 
manner,  be  obtained  theoretically  by  substituting  in  the  equation 

(330) 

for  D'  the  density  of  the  gas  referred  to  that  of  air  as  unity,  and 
for  p'  the  value  of  the  pressure  in  terms  of  the  barometric  height, 
y  being  taken  as  1.41 ;  or  it  may  be  obtained  more  simply  by  di- 
viding   

V  =  332.3  m.  x  Vl  +  «0  (231) 

by  the  square  root  of  the  density  of  the  gas  referred  to  air  as  unity. 


RELATING    TO    SOUND    AND    LIGHT. 


119 


At  zero  degrees  Centigrade,  we  have  for  the  theoretical  value  of  the 
velocity  of  sound  in  the  following  gases : 


Air,      ....         ...    332 

Hydrogen, 1269 

Oxygen, 317 


Carbon  dioxide,  ....  262 
Carbon  monoxide,  .  .  .  337 
Olefiantgas, 314 


196.  Velocity  of  Sound  in  Air  and  other  Gases,  as 
affected  by  their  not  beiny  Perfect  Gases.  The  for- 
mulae of  Mariotte,  Charles,  and  Poisson  are  only  applicable  to 
perfect  gases.  This  condition  requires  the  elasticities  to  be  perfect, 
and  the  excess  of  the  elastic  force  which  gives  rise  to  wave  propaga- 
tion to  be  indefinitely  small  when  compared  with  the  elasticity  of 
the  gas  in  its  quiescent  state. 

A  series  of  experiments  made  by  Eegnault,  the  results  of  which 
are  given  in  the  Comptes  Eendus,  Vol.  66,  page  209,  show  that 
these  conditions  are  not  fulfilled,  and  that  the  theoretical  velocity 
therefore  differs  from  the  actual.  The  sounds  were  made  in  tubes 
of  different  cross-section,  by  discharging  a  pistol  with  different 
charges  of  powder.  the  results  are  grouped  in  the  following  table : 


Diameter  of  Tube,  0.108  m. 

Diameter  of  Tube,  0.3  m. 

Pi  am.  of  Tube, 

Length,  566.74  m. 

Length,  1905  m. 

1.10  m. 

Charge,  0.3  gr. 

Charge,  0.4  gr. 

Charge,  0.3  gr. 

Ch'ge, 
0.4  gr. 

Charge,  1.5gr. 

Charge,  l.OOgr. 

Dis- 
tances. 

?*Tean 
Veloc- 
ities. 

Dis- 
tances. 

Mean 
Veloc- 
ities. 

Dis- 
tances. 

Mean 
Veloc- 
ities. 

Mean 
Veloc- 
ities. 

Dis- 
tances. 

Mean 
Veloc- 
ities. 

Dis- 
tances. 

Mean 
Veloc- 
ities. 

566.74 

330.99 

1351.95 

329.95 

1905 

asi.91 

332.37 

3810.3 

332.18 

749.1 

334.16 

1133.48 

328.77 

2703.00 

328.20 

3810 

328.72 

330.34 

7620.6 

330.43 

9201 

333.20 

1700.22 

328.21 

4055.85 

326.77 

11430.0 

329.64 

1417.9 

332.50 

2266.96 

327.04 

5407.80 

*323.34 

15240.0 

328.96 

2835.8 

331.72 

2833.70 

327.52 

5671.8 

asi.24 

8507.7 

330.87 

*  Un- 

11343.6 

330.68 

certain. 

14179.5 

330.56 

17015.4 

330.50 

19851.3 

330.52 

197.  From  these  results  we  see  :  1°,  that  the  mean  velocity  of 
the  same  wave  decreases  from  the  origin  ;  2°,  that  it  is  less  for  the 
same  charge  and  route  in  tubes  of  smaller  diameter ;  3°,  that  it 


120  ELEMENTS    OF    WAVE    MOTION, 

decrease  less  rapidly  in  tubes  of  larger  diameter.  Regnault  also,  by 
means  of  sensitive  diaphragms,  followed  the  course  of  the  waves 
after  they  became  inaudible,  and  obtained  similar  results  with  re- 
spect to  these.  He  found  that  a  sound  produced  by  a  pistol  dis- 
charge, of  one  gramme  of  powder,  became  inaudible  at  distances  of 
1150,  3810,  and  9540  metres,  in  tubes  of  0.108  m.,  0.30  m.,  and 
1.10  m.  diameter,  respectively,  and  that  the  waves  became  insensible 
after  traveling  distances  of  4056,  11430,  and  19851  metres  respec- 
tively. In  the  tube  of  1.1  m.  diameter,  with  a  charge  of  2.4  grains, 
the  wave  ceased  to  be  audible  at  58641  metres,  and  ultimately  ceased 
at  97735  metres.  These  distances  of  audibility  are,  approximately, 
directly  proportional  to  the  diameters  of  the  tube. 

198.  The  mathematical  theory  discusses  the  case  of  a  perfect 
gas,  and  assumes  that  the  propagation  in  an  indefinite  tube  is  con- 
tinuous.    The  above  experiments  show  that  this  is  not  really  the 
case.    The  assumptions  made  by  implication  in  a  perfect  gas  are : 

1°.  That  the  laws  of  Mariotte,  Charles,  and  Poisson  are  true, 
but  it  is  well  known  that  no  gas  obeys  exactly  these  laws. 

2°.  That  its  elasticity  is  unaffected  by  admixture  with  other 
gases. 

3°.  That  the  gas  offers  no  opposition  by  its  inertia  to  wave 
transmission ;  but  experiment  shows  that  an  intense  disturbance 
always  produces  a  real  motion  of  the  surrounding  particles,  which 
increases  the  velocity,  especially  within  sensible  distances  from  the 
origin.  Such  is  the  case,  no  doubt,  in  cannon  discharges,  violent 
lightning-flashes,  and  other  like  instances. 

4°.  Theory  supposes  the  excess  of  pressure  due  to  a  vibrating 
body  small,  in  comparison  with  the  quiescent  barometric  pressure ; 
but  in  the  cases  cited  above,  the  excess  of  pressure  at  Mie  origin 
may  be  large,  and  hence  cause  an  increase  in  the  value  of  V  near 
the  origin.  Therefore,  the  correction  of  Art.  193,  called  that  of 
La  Place,  in  such  cases  is  not  exact. 

199.  Regnault  ascribes  as  the  principal  cause  of  the  diminution 
of  the  intensity,  the  loss  of  kinetic  energy  by  the  reaction  of  the 
sides  and  ends  of  the  tube,  and  confirms; this  by  the  fact  that  the 
sounds  are  quite  audible  outside  the  tube  during  their  first  passage, 
and  in  a  less  degree  at  veach  succeeding  passage.     As  a  secondary 
cause,  he  ascribes  the  influence  of  the  walls  of  the  tube  in  dimin- 
ishing the  elasticity  without  affecting  the  density.     This  is  con- 


RELATING    TO    SOUND    AND    LIGHT.  121 

firmed  by  the  fact  that  in  the  above  experiments,  where  the  waves 
have  been  produced  by  the  same  charge,  and  hence  have*  the  same 
sensibility  at  the  origin,  they  have  not  the  same  intensity  after 
traveling  over  equal  routes.  The  mean  limiting  velocity  ought, 
therefore,  to  be  the  same,  if  the  weakening  is  due  to  the  loss  of  mv2 
on  account  of  the  sides.  The  experiments  show  that  this  is  not  the 
case ;  hence,  the  sides  exercise  another  effect  on  air  different  from 
that  indicated  as  the  principal  cause  of  the  diminution  of  the  in- 
tensity, an  action  affecting  the  elasticity  and  not  the  density.  In 
free  air  this  effect  would  be  null,  and  in  the  tube  of  1.1  m.  it  is 
taken  as  approximately  so.  The  mean  velocity  of  propagation,  in 
dry  air  at  0°  C.,  of  a  wave  produced  by  the  discharge  of  a  pistol, 
and  estimated  from  the  origin  to  the  point  at  which  its  sensibility 
can  no  longer  be  appreciated  by  the  ear  is,  according  to  Regnault's 
experiments, 

F  =  330.6  m. 

The  mean  limiting  velocity,  considered  from  the  origin  to  the 
point  at  which  its  existence  can  110  longer  be, detected  upon  a  sen- 
sitive diaphragm,  is 

V  =  330.3  m., 

which  differs  from  the  mean  limiting  velocity  in  the  1.1  m.  tube  by 
only  0. 32  m. 

200.  Velocity  of  Sound  in  Gases  independent  of 
the  Barometric  Pressure.  Since  an  increase  in  the  baro- 
metric pressure  increases  the  elasticity  and  density  in  the  same 
proportion,  theory  indicates  that  no  change,  due  to  this  cause  alone, 
will  take  place  in  the  velocity.  The  experiments  of  Stampfer  arid 
Myrbach  in  the  Tyrol,  in  1822,  between  two  stations  whose  differ- 
ence in  altitude  was  1364  m.,  and  of  Bravais  and  Martins  in  Swit- 
zerland, in  1844,  between  two  stations  whose  difference  of  level  was 
2079  m.,  indicated  no  variation  in  the  velocity,  due  to  the  change 
in  the  barometric  pressure.  Regnault's  experiments  upon  air  in 
the  tube  0.108  m.  in  diameter,  over  a  distance  of  567.4  m.,  with 
pressures  varying  from  0.557  m.  to  0.838  m.,  and  over  a  distance  of 
70.5  m.,  with  pressures  varying  from  0.247  in.  to  1.267  m.,  found 
no  variation  in  the  velocity,  due  to  this  cause. 

The  theoretical  ratio  of  the  velocities  of  sound  in  gases,  given  by 


122  ELEMENTS    OF    WAVE    MOTION. 


V  _        I'D 
T~  ~  \  ~D" 


was  experimentally  confirmed  to  a  near  degree  of  approximation  in 
the  cases  of  hydrogen,  carbon  dioxide,  and  air.  The  tube  0.108  m., 
filled  for  a  length  of  567.4  m.,  gave  for  hydrogen  3.801  m.,  for  car- 
bon dioxide  0.7848  m.,  which  differ  but  little  from  the  theoretical 
values  3.682  m.  and  0.8087  m.,  the  velocity  in  air  being  taken  as 
unity.  Hence  the  formula  may  be  taken  as  an  expression  for  the 
limiting  law.  The  determination  of  the  velocity  of  sound  in  free 
air  was  made  by  means  of  reciprocal  cannon  discharges.  There 
were  two  series  of  these  experiments.  For  the  first,  consisting  of  18 
discharges,  the  membrane  being  1280  metres  distant,  the  mean 
velocity,  referred  to  dry  air  at  0°  C.,  was  found  to  be 

V  =  331.37  m. 

For  the  second  series,  of  149  discharges,  over  a  distance  of 
2445  m.,  during  11  days  of  trial,  with  the  temperature  of  the  air 
varying  from  1.5°  to  21.8°  0.,  and  with  great  variations  in  the 
wind,  the  mean  velocity,  referred  to  dry  air  at  0°  C.,  was 

V  =  330.7  m., 
a  sensible  diminution  of  the  velocity,  due  to  the  increased  distance. 

201.  Velocity  of  Sound  in  Liquids.  The  value  of  the 
velocity  of  sound  in  liquids  is  likewise  given  by  the  general  formula 

=  V  ~D^    ~  V  ~'^~  X  ~D 

(233) 


in  which  H  is  the  arbitrary  barometric  height,  dm  the  density  of 
mercury,  and  g  the  acceleration  due  to  gravity.  The  numerator  is 
then  the  pressure  due  to  the  height  of  the  barometer,  and  when 
divided  by  A,  which  is  the  diminution  of  the  volume  due  to  the 
increase  of  pressure,  gdmH  gives  the  ratio  of  the  pressure  to  the 
corresponding  compression,  and  is  therefore  the  measure  of  the  elas- 
tic force  of  the  medium.  The  square  root  of  this  quantity,  divided 


RELATING    TO    SOUND    AND    LIGHT.  123 

l>y  the  square  root  of  the  density,  will  be  the  value  of  the  velocity 
of  sound  in  the  liquid. 

202.  Colladon  and  Sturm  made  a  series  of  experiments  to  de- 
termine the  actual  value  of  the  velocity  of  sound  in  water,  in  Lake 
-Geneva,  in  the  year  1826.  The  sound  was  caused  by  the  strokes  of 
a  hammer  upon  a  bell  submerged  one  metre  below  the  surface,  and 
so  arranged  that  the  epoch  of  the  stroke  could  be  determined  by  a 
flash  of  powder.  The  instant  of  hearing  the  sound  was  indicated 
by  a  stop-watch  to  within  one-quarter  of  a  second.  The  distance 
traveled  by  the  sound  was  found  to  be  13487  m.  to  within  20  m., 
and  the  time  of  this  travel,  from  a  mean  of  many  experiments,  was 
found  to  be  9.4  s.  The  temperature  of  the  water  was  8.1°  C.,  its 
•density  at  that  temperature,  referred  to  that  of  water  at  the  stand- 
ard temperature,  was  unity  plus  a  negligible  fraction,  its  compressi- 
bility was  taken  at  .0000495,  and  the  barometric  height  at  76  cm. 
The  density  of  mercury  referred  to  the  same  temperature  is  13.544, 
and  g  —  9.8088. 

Making  these  substitutions  in  the  preceding  formula,  we  find 


/9. 

=  V~ 


9.8088  x  13.544  x76 


006645S 


The  actual  velocity  found  was  -  -  =  1435  m.,  differing 

y.4 

from  the  theoretical  value  but  7  m.  The  latter  may  itself  vary 
within  wider  limits,  on  account  of  the  inexactness  of  the  value  of 
the  compressibility  of  water,  whose  most  probably  correct  value, 
from  the  experiments  of  Regnault,  is  assumed  to  be  .00004685. 

203.  The  principal  facts  derived  from  these  experiments  of 
Colladon  are  (Tome  XXXVI,  Annales  de  Chimie)  that  at  distances 
beyond  200  metres  the  quality  of  the  sound  is  changed,  and  the 
sensation  is  similar  to  the  quick,  brief  noise  'produced  by  the  strik- 
ing together  of  two  knife-blades  in  air.  The  diminution  of  inten- 
sity with  the  distance  is  noticed,  and  at  short  distances,  greater 
than  200  metres,  it  is  not  possible  to  tell  whether  the  sound  origi- 
nates at  a  near  origin  of  weak  intensity,  or  at  a  distant  origin  with 
increased  intensity.  The  duration  is  less  than  in  air  ;  as  it  should 

be  from  its  value  -,  A  being  greater  and  V  being  smaller  in  air 


124:  ELEMENTS    OF    WAVE    MOTION. 

than  in  water.  When  the  vibrations  proceeding  from  the  sounding 
body  reach  the  surface  of  the  water  at  great  angles  of  incidence,  the 
sound  does  not  pass  into  the  air.  At  distances  greater  than  400  to 
500  metres,  the  ear  in  air  does  not  hear  the  sound  originating  in 
the  water.  At  200  metres  the  sound  is  readily  heard.  In  these 
experiments,  the  bell  being  placed  2  metres  below  the  surface,  the 
angle  of  incidence  at  400  metres  is  approximately  89°  43'  ;  at  200 
metres,  89°  26'. 

Finally,  the  existence  of  a  sharper  acoustic  shadow  shows  that 
the  wave  lengths  are  proportionally  shortened  in  water  compared 
with  the  waves  made  in  air  by  the  same  sounding  body. 

204.  Velocity  of  Sound  in  Solids.     The  ordinary  solids 
upon  which  experiments  have  been  made  for  the  determination  of 
the  velocity  of  sound  are  glass,  the  various  metals,  and  wood.     In 
the  latter,  from  the  manner  of  its  growth  in  the  tree,  the  three  di- 
rections, along  the  axis,  in  the  direction  of  the  radius,  and  normal 
to  the  plane  of  these  two,  possess  necessarily  different  elasticities. 
The  coefficients  of  elasticity  also  differ  in  different  species,  and  in 
the  same  species,  when  grown  in  different  localities,  under  different 
circumstances   of   soil,   temperature,    and    moisture.      Reasonably 
exact  determinations  belong  then  only  to  the  particular  specimen 
experimented  upon,  and  mean  values  are  usually  taken  for  any  one 
kind  of  wood  in  a  given  direction.     In  metals  and  glass,  variations 
of  the  coefficients  arise  from  the  methods  of  their  manufacture,  and 
modifications  result  from  every  circumstance  which  affects   their 
density  and  other  physical  properties.     None  of  the  solids  can  be 
said  to  be  perfectly  homogeneous  ;  but  on  the  assumption  that  they 
are  approximately  so,  different  experimenters  have  obtained  values 
for  their  coefficients  which  do  not  vary  between  very  wide  limits. 

205.  In  solids,  the  sound  may  result  either  from  transversal  or 
from  longitudinal  vibrations.     In  the  cases  here  considered,  the 
vibrations  are  understood  to  be  longitudinal,  that  is,  the  molecular 
displacements  are  in  the  direction  of  the  propagation. 

When  a  solid  bar,  taken  as  homogeneous,  transmits  a  longitudi- 
nal vibration,  the  velocity  of  the  propagation  has  been  found  to  be 
given  by  the  equation 

(234) 


•VI- 


RELATING    TO    SOUND    AND    LIGHT. 


125 


Substi- 


in which  A  is  the  elongation  due  to  the  weight  of  the  bar. 
tuting  for  A  its  value  in  terms  of  Young's  modulus, 


and  making  s  equal  to  one  square  centimetre,  I  equal  to  one  metre, 
and  P  the  weight  of  the  bar,  we  have 


(236) 


r=t/5' 


the  same  in  form  as  has  been  found  for  gases  and  liquids. 

206.  Different  methods  have  been  employed  to  find  E9  viz.,  by 
the  direct  method  of  elongations  or  compressions,  by  flexure,  by 
transversal  and  by  torsional  vibrations   of  the  bar.     The  values 
given  for  the  different  metals,  in  Art.  23,  have  been  obtained  by 
Wertheim,  by  the  method  of  elongations.     Could  we   accurately 
determine  the  velocity  of  sound  in  solids  by  direct  experiment,  the 
value  of  E  could  be  readily  found  by  the  solution  of  the  above 
equation.     But  this  velocity  being  very  great  compared  with  that  in 
air,  and  because  of  the  impracticability  of  finding  sufficiently  long 
homogeneous  lengths,  an  accurate  determination  of  E  by  this  means 
is  impossible.     Biot,  by  a  direct  experiment  on  951  metres  in  length 
of  cast-iron  pipe,  found  that  the  velocity  was  10.5  times  that  in  air; 
but  the  want  of  homogeneity,  due  to  the  numerous  leaded  joints, 
without  doubt  influenced  this  result  appreciably.     Wertheim  found 
about  the  same  value  in  wrought  iron,  by  experimenting  upon 
4067.2  metres  of  telegraph  wire. 

207.  Assuming  the  experimental  values  for  E  given  in  Art.  23, 
and  taking  g  to  be  981  dynes,  the  velocities  of  sound  are,  by  the 
above  formula,  found  to  be  as  follows  : 


E. 

D. 

F  IN  CENTIMETRES. 

RATIO  TO 
FIN  AIB. 

Lead,     .    . 

177x981xl06 

11.4 

1.23  xlO5 

3.7 

Gold,     .    . 

813  x  981  x  106 

19.0 

1.74x108 

5.3 

Silver,   .    . 

736  x  981  x  106 

10.5 

2.61xl05 

8.0 

Copper,  .    . 

1245  x  981  x  106 

8.6 

3.56xl05 

10.7 

Iron,      .    . 

1861  x  981  x  106 

7.0 

5.13xl05 

15.5 

Steel,     .    . 

1955  x  981  x  106 

7.8 

4.99xl05 

15.0 

126 


ELEMENTS    OF    WAVE    MOTION. 


For  glass,  with  density  of  2.94,  V  has  been  found  to  be,  by  the 
same  method,  4.53  cm.  x  105 ;  and  for  brass,  of  density  of  8.47, 
V  =  3.56  cm.  x  105;  or  ,13.6  and  10.8  times  the  velocity  in  air, 
respectively. 

208.  The  following  velocities  of  sound  in  wood,  deduced  from 
the  observations  of  Wertheim  and  Ohevandier  (Comptes  Rendus, 
1846),  are  taken  from  "Everett's  Physical  Constants,"  page  65,. 
from  which  also  several  of  the  above  numbers  have  been  obtained : 


ALONG  FIBRE. 

RADIAL. 

TANGENTIAL. 

Pine,     

3.32xl05 

2.  83  xlO5 

1.59xl05 

Beech,  .    .    . 

3.34  xlO5 

3.67xl05 

2.83xl05 

Birch, 

4.  42  xlO5 

2.14xl05 

3.03xl05 

Fir,  

4.64xl05 

2.  67  XlO5 

1.57  xlO5 

209.  The  preceding  values  of  the  velocities  of  sound  in  solids 
are  true  only  when  the  medium  is  in  the  form  of  a  bar  of  small 
cross-section.  Wertheim  has  shown  by  his  investigations,  based  on 
the  theory  of  Cauchy,  that  the  corresponding  velocities  in  extended 
homogeneous  solids  are  greater  than  the  above  results  in  the  ratio  of 


210.   Reflection    and    Refraction  of  Sound.     The 

laws  deduced  in  Art.  77  for  the  reflection  and  refraction  of  wave 
motion  are  applicable  to  the  undulations  of  sound.  From  the 
equation 

sin  0  =  |»  sin  0',  (237) 

the  direction  of  any  deviated  ray  or  that  of  any  deviated  plane  wave 

Y 
by  a  plane  surface,  can  be  found  when  -=7  is  substituted  for  \i.    If 

V  >  V,  then  0'  >  0,  and  the  refracted  ray  is  thrown  from  the 
normal ;  conversely,  if  V  <  V,  then  0'  <  0,  and  the  refracted 
ray  is  bent  towards  the  normal.  A  ray  of  sound  in  air,  incident  on 
the  surface  of  water,  will  be  refracted,  provided  the  angle  of  inci- 
dence be  less  than  13°  26' ;  for  since  V  in  water  is  about  1428  m.f 
and  V  in  air  about  332,  we  have 


RELATING    TO    SOUND    AND    LIGHT.  127 


and  sin  0  =  .2325,        or        (f>  =  13°  26'. 

For  greater  incidences  the  ray  is  totally  reflected,  and  does  not 
enter  the  water. 

211.    Consequences  of  the  Laws  of  Reflection. 

1°.  If  a  sound  originate  at  one  of  the  foci  of  an  ellipsoid,  it  will 
be  reflected  to  the  other  focus. 

2°.  If  at  the  focus  of  a  paraboloid,  the  rays  of  sound  will  be 
reflected  in  lines  parallel  to  the  axis,  and  can  be  again  collected  at 
the  focus  of  another  similar  paraboloid,  with  sensibly  undiminished 
intensity.  The  slightest  sound,  as  the  ticking  of  a  watch,  may  be 
employed  to  illustrate  this  case  of  reflection. 

3°.  The  speaking-trumpet  and  speaking-tube  are  employed  to 
prevent  the  too  rapid  dissipation  of  sound.  The  former,  partly  by 
reflection  from  its  sides  and  largely  by  resonance,  concentrates  the 
sound  within  the  volume  of  the  cone  whose  apex  is  the  mouth- 
piece and  whose  section  is  that  of  the  other  end  of  the  trumpet. 
The  speaking-tube  confines  the  energy  in  the  narrow  compass  of 
the  tube,  the  loss  being  insignificant  in  the  ordinary  lengths  em- 
ployed. 

4°.  When  a  sound  is  reflected  by  any 
obstacle  which  prevents  its  direct  trans- 
mission, and  the  observer  is  at  such  a  dis- 
tance that  the  direct  and  reflected  sounds 
are  not  confounded,  the  reflected  sound  is 
called  an  echo.  Thus,  if  A  be  the  position 
of  the  observer,  S  the  origin  from  which  a  Figure  28. 

sound  of  short  duration  emanates,  and  W 
the  obstacle,  such  as  a  wall,  then  the  direct  sound  will  reach  the 

Q   A 

observer  in  the  time  ^-j,  and  the  reflected  sound  in  the  time 
oo&A 

SW  +  WA  .       A0         r£  SW  +  WA  —  SA  , 

-3^-4  —  ,  the  temperature  being  0°  0.     If  -      —^  ^         -  be 

sufficiently  great,  so  that  the  reflected  sound  arrives  after  the  ces- 
sation of  the  direct  sound,  then  the  echo  will  be  heard,  provided 
the  intensity  be  of  sufficient  value.  If  the  two  sounds  commingle, 


128  ELEMENTS    OF    WAVE    MOTION. 

the  resultant  sound  will  be  prolonged,  and  partial  resonance  will 
ensue.  The  number  of  distinct  impressions  distinguished  by  the 
ear  will  determine  the  shortest  difference  of  route  necessary  to  es- 

332 
tablish  the  echo.     Thus,  if  we  take  nine  per  second,  — —  or  37  m. 

is  the  shortest  difference  of  route  at  0°  C. 

5°.  The  conditions  of  interference  of  sound  are  the  same  as 
those  discussed  in  Arts.  65-68.  Hence,  it  is  theoretically  possible 
that  two  sounds  affecting  the  ear  simultaneously  will  result  in 
silence,  and  practically  it  will  be  shown,  in  the  lectures  on  this  part 
of  the  course,  that  such  an  experiment  is  also  possible.  Other  illus- 
trations of  interference  are  also  reserved  for  the  lectures. 

212.  Refraction  of  Sound.    In  order  that  the  rays  of 
sound  shall  converge  after  deviation  by  refraction,  we  see  from  the 

Y 
formula  that  fi  —  -=-,   must  be   greater  than   unity.     Then  the 

deviated  wave  will,  in  general,  become  converging,  and  the  energy 
accumulate  on  an  ever  decreasing  surface.  Examining  the  table, 
Art.  195,  we  see  that  V  in  carbon  dioxide  is  262  m.,  and  hence, 
when  the  incident  medium  is  air, 

-  332  -  1  25 
^    -262-    °5' 

and  sin  $  =  1.25  sin  </>'.  (238) 

The  sound  lens  devised  by  Sondhaus  is  a  double  convex  lens  of 
collodion  filled  with  carbon  dioxide,  which  collects  the  sound  rays 
proceeding  from  any  sonorous  body  and  concentrates  them  appre- 
ciably at  another  point  on  the  opposite  side  of  the  lens.  By  means 
of  a  concave  lens  of  the  same  material,  filled  with  hydrogen, 
V  =  1269  m.,  it  will  be  evident,  after  the  study  of  the  properties 
of  lenses,  as  explained  in  optics,  that  a  similar  result  would  be 
effected.  The  slight  noise  produced  by  the  ticking  of  a  watch  may 
be  collected  by  this  means  at  a  point  so  that  the  noise  is  audible, 
when  without  this  assistance  it  would  be  inappreciable  at  the  same 
point. 

213.  General  Equations  for  the  Vibratory   Mo- 
tion of  a  Stretched  String.     The  bodies  usually  employed 
to  produce  musical  sounds  by  their  vibrations  are  strings,  rods  air- 


RELATING    TO    SOUND    AND    LIGHT.  129 

columns,  plates,  bells,  etc.  When  the  vibrations  of  the  particles 
are  perpendicular  to  the  direction  of  wave  propagation,  they  are 
called  transversal,  and  when  in  the  same  direction,  longitudinal. 

We  will  first  consider  the  vibrations  of  a  perfectly  elastic  and 
flexible  string,  supposed  to  be  stretched  between  two  points  whose 
distance  apart  is  I,  by  a  force  which  produces  a  tension  T.  Let  the 
elongation  be  that  given  by 

I' -1  =  ^1,  (239) 

in  which  I  is  the  natural  length,  I'  the  length  after  the  tension  T  is 
applied,  and  E  is  the  longitudinal  modulus.  If  the  displacements 
of  the  string  fro  in  its  position  of  rest  be  due  to  the  incessant  action 
of  forces  whose  rectangular  accelerations  are  X,  Y,  Z,  these  with 
the  tension  T  will  be  the  only  extraneous  forces  considered. 

Let  m  be  the  mass  of  any  element  \  x,  y,  z,  x  +  dx,  y  -f  dy, 
z  -f  dz,  the  co-ordinates  of  its  extremities  and  its  length  ds ;  cc  the 
area  of  its  cross-section,  and  p  its  density ;  then 

m  =  pads. 
Let  the  components  of  T  at  x,  y,  z,  be 

T—          T^-          T-- 
ds  9  ds '  ds' 

and  at  x  -f-  dx,  y  +  dy,  z  -f-  dz,  be 

2*  ~  4-^7*—  T^y.   \   dT^-  77^4_/77T^J. 

ds  ds'  ds  ds9  ds  ds' 

The  general  equations  of  motion  will  then  be 


(240) 


214.   These  equations  are  simplified  when  we  suppose  that  the 
string  is  arbitrarily  displaced  from  its  position  of  equilibrium,  and 
9 


130 


ELEMENTS    OF    WAVE    MOTION. 


abandoned  to  itself,  without  the  action  of  the  forces  Jf,  Y,  Z.  It 
will  then  oscillate  about  its  position  of  rest,  and  the  only  extraneous 
force  that  acts  will  be  the  tension  T,  whose  intensity  will  vary  be- 
tween known  limits.  Let  the  axis  of  x  coincide  with  the  string  in 
its  position  of  rest,  and  the  co-ordinates  of  the  element  m,  at  the 
time  t,  be  x  -f- 1,  77,  £  If  the  displacement  be  supposed  small,  £, 
??,  and  £  are  functions  of  x  and  t,  and  x  is  independent  of  t,  and  the 
above  equations  reduce  to 


(241) 


ds 


Let  T'  be  the  tension  when  the  string  is  straight,  and 
the  string  is  displaced;  the  length  of  the  element  is  in  the  first 
case  dx,  and  in  the  second  ds  ;  these  are  connected  by  the  equations 


(242) 
(243) 

(244) 

(245) 


ds  =  a 

d&  =  (dx  +  d!;)*  +  drf  +  dt? ; 
from  which,  when  dr)  and  d£  are  very  small,  we  have 
ds  =  dx  -\-  d%, 


T=  T'  +  E- 
dx 

Substituting  in  Eqs.  (241),  we  have 


(246) 


RELATING    TO    SOUND    AND    LIGHT.  131 

E  T' 

Keplacing  —  and  —  by  u2  and  ^  respectively,  we  have 
pa  pa 


_ 

dt*  ~  da?-' 

^n  _  2dfy 

dP  ~  do?' 

*£-  v*d-S. 

dP  ~  dx* 


(247) 


The  integration  of  these  three  partial  differential  equations  give 
(Analytical  Mechanics,  Appendix  IV), 


-  vt),  (248) 

?  =  F(x+  vt)  +f(x-vt).  ) 

215.  The  first  equation  determines  the  longitudinal  vibrations, 
or  those  along  the  axis  of  the  string,  and  the  other  two  give  the 
transversal  vibrations  along  y  and  z  respectively.     Because  of  the 
independence  of  the  differential  equations,  the  three  vibrations  in 
general  coexist  and  are  wholly  independent  of  each  other,  and  since 
the  differential  equations  are  of  the  same  form,  we  see  that  the  two 
kinds  of  vibrations  are  subjected  to  the  same  laws.    They  may  each 
be  discussed  separately.     Each  is  due  to  a  progressive  motion  for- 
ward and  backward  along  the  string.     These  motions  may  be  of 
the  most  varied  character,  but  the  particular  form  of  the  motion 
depends  on  the  form  of  the  functions  whose  symbols  are  F  and/. 
The  only  conditions  imposed  so  far  are  that  for  x  =  0  and  x  =  I, 
I,  TJ,  and  £  are  zero  for  all  values  of  t.     These,  together  with  any 
assumed  initial  conditions,  will  enable  us  to  determine  the  form  of 
the  functions   F  and  /,  and  thus  complete  the  solution  of  the 
problem. 

216.  Since  the  vibrations  parallel  to  y  and  z  are  exactly  alike  in 
every  particular,  the  discussion  of  one  will  do  for  the  other,  and  we 
will  consider  that  of  y,  given  by  the  equation 

il  =  F(x  +  vt)+f(x-  vt}.  (249) 


132  ELEMENTS    OF    WAVE    MOTION. 

Assume  the  conditions"  that  at  the  epoch,  or  when  t  —  0, 

??  =  0(z)         and         jj=vil>'(x),  (250) 

in  which  the  functions  0  and  0  are  supposed  known,  and  that  0'  is 
the  derived  function  of  -0.    If  t  =  0,  Ave  have 


7?  =  0(3)  =  ^(z)+/»,  (251) 

J-J«-y<»)-=^-<4-/f(*);/  (252) 

/.    1>(x)  =  F(x)-f(xj;  (253) 

and  hence,                         F(x)=+M+W9  (354) 


Therefore  ^(z)  and  /(#)  are  known  for  all  values  of  x  from 
0  to  I,  when,  as  is  supposed,  0  (x)  and  -0  (x)  are  known  between 
the  same  limits. 

For  the  extremities,  we  have,  by  placing  x  =  0  and  x  =  I, 

F(vt)  +f(-vt)  =  0;  (256) 

F(l  +  vt)+f(l-vt)  =  0;  (257) 

whence,  F  (vt)  and.f(—vt)  are  equal,  with   contrary  signs,  and 
thus  become  known  for  all  values  from  t  =  0  to  t  =  <x>  . 

217.  The  value  of  r\  can  be  expressed  by  means  of  a  single  func- 
tion by  substituting  vt  +  I  —  x  for  vt  in  Eq.  (257)  ;  whence, 

-F(M  —  x  +  vt)  =f(x  —  vt)  ;  (258) 

which  in  Eq.  (249)  gives 

?i  —  F(x  +  vt)  —  F  (21  —  x  +  vt).  (259) 

Again,  for  vt,  in  Eq.  (257),  substitute  I  +  vt  ;  then 

.F  (2?  +  vt)  =  —f(—vt)  =  F  (vt)  ;  (260) 

whence  we  conclude  that  the  function  F  takes  the  same  value  when 
the  variable  vt  is  increased  by  21  ;  and  therefore  by  4/,  Ql,  SI,  ---- 


RELATING    TO    SOUND    AND    LIGHT. 


133 


or  2nl,  n  being  a  positive  whole  number.  Therefore,  if  F  (vt)  is 
known  from  vt  =  0  to  vt  =  21,  its  value  is  known  for  all  values 
from  t  =  0  to  t  =  oo . 

Replace  vt  by  I  —  vt,  in  Eq.  (257),  vt  being  less  than  I ;  then 


F(2l-vt)  =  - 


(261) 


but  f(vt)  is  known  for  all  values  of  vt  between  0  and  Z;  therefore 
F  (vt)  is  known  for  all  values  of  vt  between  I  and  21. 

Hence,  the  value  of  F  (x  4-  vt)  is  known  for  all  values  of  x-\-vt 
from  0  to  oo ;  and,  similarly,  the  value  of  f(x  —  vt)  can  be  found 
for  all  values  between  0  and  —  oo;  and  therefore  the  problem  is 
completely  solved. 


\A, 


Figure  29, 

218.   The  function  whose  symbol  is  FIB  subject  to  the  following 
conditions,  derived  from  Eqs.  (256),  (257),  (260),  (261), 


F(x)  =  -F(-x), 
F(l  +  x)  =  -F(l-x), 
F  (x)  =  F  (21  +  x)  =  F  (±1  +  x)  — 


F(x]  =  -F(2l-x)  =  -F(±l-x)  = 


(362) 
(263) 

(264) 
(265) 


From  Eq.  (262)  we  see  that  the  curve  represented  by  77  =  F  (x) 
is  continued  in  similar  forms  on  each  side  of  0  in  the  figure  ;  from 
Eq.  (263),  that  the  forms  are  similar  on  each  side  of  A;  from  Eq. 
(264),  that  the  form  is  repeated  from  0'  to  0"  exactly  as  from  0  to 
0';  and  from  Eq.  (265),  that  the  form  of  the  curve  inverted  is  the 
same  from  0'  to  A  as  from  0  to  A. 

The  motion  of  any  particle  is  that  of  oscillation  about  its  place 

21 
of  rest,  and  of  which  the  period  is  —     This  vibratory  motion  is 

gradually  diminished,  while  the  period  remains  unchanged,  because 


134  ELEMENTS    OF    WAVE    MOTION. 

of  the  energy  communicated  to  the  air,  and  through  the  points  of 
attachment  to  other  bodies.     The  time  of  one  complete  oscillation  is 


(366) 


and  the  number  of  oscillations  in  the  unit  of  time  is 


Therefore  in  the  transversal  vibrations  of  a  string,  the  resulting 
pitch  is  inversely  proportional  to  its  length,  directly  as  the  square 
root  of  the  tension  when  straight,  and  inversely  as  the  square  root 
of  the  density  by  the  area  of  cross-section. 

219.  The  number  of  longitudinal  oscillations  in  the  unit  of 
time  is 


whence  the  pitch  depends  only  upon  the  length  of  the  string  and 
the  material  of  which  it  is  made,  and  is  independent  of  the  tension, 
unless  the  latter  should  be  so  considerable  as  to  change  the  value 
of  E.  Experiment  appears  to  indicate  that  the  longitudinal  pitch 
increases  slightly  with  the  tension  ;  but  this  may  be  accounted  for 
in  the  elongation  experienced,  which  is  always  accompanied  with  a 
slight  diminution  of  density  p,  and  should  this  occur,  the  formula 
indicates  that  the  pitch  should  rise. 

The  ratio  of  the  numbers  for  the  same  string  is  given  by 

^.;:'V  (269) 

M.  Cagniard  Latour  experimented  on  a  cord  of  14.8  m.  in 
length,  and  found 


and        AZ  =  0.05  m. 

Substituting  in  the  formula,  we  have 
188 


whence,  A?  =  0.052  m.,  a  sufficiently  near  approximation. 


RELATING    TO    SOUND    AND    LIGHT.  135 

220.  The  preceding  values  of  n  and  ri  are  the  least  numbers  of 
transversal  and  longitudinal  vibrations  of  the  string,  and  therefore 
correspond  to  its  fundamental  tones;  but  we  know  that  each  of  the 
vibrations  is  decomposed  into  any  number  of  vibrations  of  equal 
periodicity,  when  the  string  is  divided  into  a  like  number  of  sym- 
metrical parts.  This  can  be  shown  more  readily  when  the  integral 
equation  is  expressed  in  a  series  which  is  a  function  of  sines  and 
cosines.  Thus,  it  is  evident  that  a  possible  solution  of  the  differen- 
tial equation 


is  given  by 

%-nvt        n     .     invt\    .    inx 
cos  —  —  j-  BI  sm  --    sm  --  ,  (271) 


( 


when  the  conditions  with  respect  to  the  extreme  points  are  un- 
changed. In  this  equation,  i  is  any  entire  positive  number  which 
marks  the  order  of  the  term,  and  Air  B^  are  constant  coefficients 
depending  on  i  and  on  the  initial  state  of  the  string.  If  then  this 
state  is  such  that  77  is  constant  only  for  the  terms  for  which  i  is  a 
multiple  of  another  entire  number  n,  the  string  will  return  to  the 

21 

same  state  at  the  end  of  each  interval  of  time  — ,  which  is  the 

nv 

duration  of  its  similar  and  isochronous  vibrations.  Under  this  sup- 
position, the  n  —  1  points  of  the  curve  corresponding  to  distances 

*          -  -         -  —  t 

fi  n  n 

will  be  nodes,  that  is,  will  remain  at  rest  during  the  whole  period 
of  the  motion. 

Since  the  value  of  77  is  linear,  every  value  corresponding  to  i  = 
1,  2,  3,  4,  etc.,  will  be  a  solution,  and  the  sum  of  all  the  values  of  77 
will  also  be  a  solution  of  the  differential  equation ;  hence  we  will 
have  for  the  general  integral  equation 

.    /         j  *£7r?^£  n  •  t^lfl/l  m  1/TTtJC  tf^^i 

T)  =  SiSr\4i  cos  —7 — h  BI  sm  nrrJ  sm  ~r~*         (272) 

\  i  ill 

221.  The  values  of  A1BIJ  A2BZ,  etc.,  are  in  general  arbitrary, 
and  we  may  suppose  all  to  vanish  up  to  any  order  n,  while  the  rest 
remain  arbitrary.  If  A^B^  are  not  zero,  there  are  no  actual  nodes 


136  ELEMENTS    OF    WAVE    MOTION. 

except  the  fixed  ends,  and  the  first  simple  tone  is  that  whose  period 
is  r  and  whose  wave  length  is  21.  If  there  is  one  node,  the  period 

is  -,  and  the  gravest .  simple  tone  is  that  of  wave  length  I;  and, 

/v 

generally,   if  there  are   n  —  1   nodes,  the  period  is  - ,  and  the 

ti 

gravest  tone  is  the  (n  —  V)th  harmonic  of  the  fundamental  tone. 

When  the  string  vibrates  without  nodes,  the  series  of  harmonic 
tones  is  in  general  complete,  and  a  practised  ear  can  distinguish  ten 
or  more.  It  is  also  possible  to  make  a  string  vibrate  in  such  a  man- 
ner that  for  any  proposed  value  of  n  the  coefficients  Anttn,  A^nB^n, 
etc.,  shall  disappear,  so  that  the  component  harmonic  vibrations 

whose  periods  are  -,    — ,  etc.,  are  extinguished.    When  this  i& 

Ti          &tl 

done,  the  ear  does  not  distinguish  these  tones,  and  we  may  therefore 
conclude,  from  what  precedes,  that  each  component  tone  actually 
heard  is  produced  by  the  corresponding  harmonic  vibration  of  the 
string. 

222.  The  same  general  method  may  be  applied  to  the  longitudi- 
nal vibration  of  a  rod,  and  the  differential  equation  will  be,  as  in 
the  case  of  the  longitudinal  vibration  of  a  string,  of  the  form 

3 -"3-       ' 

of  which  the  integral  equation  is 

£  =  F  (x  +  Vt)  +  f(x  -  Vt\  (274) 

and  which  may  be  put  under  the  form  of 

..                            ITTX  I  A           iirVt        „     .     ITT  FA          /rt)VK\ 
£  =  x  +  2  cos  —j-  (Ai  cos  — j h  Bi  sin  — ^— ),       (275) 

in  which  £is  the  distance  from  the  fixed  origin  at  an}'  time  t  to  the 
particles  in  a  plane  section  of  the  rod,  of  which  the  natural  distance 
from  the  end  of  the  rod  is  x.  The  value  of  x  therefore  depends 
only  on  the  particular  section  considered,  and  is  independent  of  the 
origin  of  £;  but  if  the  vibrations  cease,  the  periodic  part  of  Eq. 
(275)  would  vanish,  and  we  would  have  £  =  x  for  all  points  of  the 
rod,  and  therefore  the  periodic  part  gives  the  displacement  (£  —  x) 
at  the  time  t  of  the  section  determined  by  the  value  of  x. 


RELATING    TO    SOUND    AND    LIGHT.  13T 

The  periodic  part  does  not  in  general  vanish  for  any  value  of  xy 
so  that  there  are  in  general  no  nodes.     But  there  will  be  n  nodes  at 

sections  for  which  x  is  any  odd  multiple  of  —  ,  provided  At,  Bi> 

vanish  for  all  values  of  i  except  odd  multiples  of  n.  Thus  the  rod 
may  have  any  number  of  nodes,  of  which  those  next  the  ends  are 
distant  from  the  ends  by  half  the  distance  between  any  two  consec- 
utive nodes. 


inx  I  '  inVt         „     .      ircVt\       /<lw/,v 

—  \Ai  cos  —  -  --  h  fy  sin  —  j—  j  ;    (276) 


223.   Differentiating  Eq.  (275),  we  have 

d%  rr  ^  .    t 

j-  =  I  —  j  Zi  sm 

which,  when  x  =  0  and  x  =  Z,  becomes 

=  1-  (277) 


But  =   , 

dx       p 

in  which  p'  is  the  natural  density,  and  p  is  the  changed  density. 
We  see,  therefore,  that  there  is  no  change  of  density  at  the  free 
ends.  If  Ai,  BI,  vanish,  except  where  i  is  a  multiple  of  n,  the 

variable  part  of  -=-  vanishes  when  #  is  a  multiple  of  -•    Hence, 
dx  n 

where  there  are  nodes,  the  sections  in  which  there  is  no  variation  of 
density  are  those  which  bisect  the  nodal  intervals  in  the  state  of 
equilibrium,  and  these  sections  of  no  variation  of  density  are  also- 

"ITTX 

sections  of  greatest  displacement,  since,  Eq.  (275),  cos  -=-  is  equal 

p 

to   ±  1  for  values  of  x  which  make  sin  —  —  0. 

l 

224.  The  vibration  represented  by  Eq.  (275)  consists  of  an  infi- 
nite number  of  simple  harmonic  vibrations,  each  of  which  might 
subsist  by  itself;  the  nth  component  would  have  n  nodes,  and  its 

21  21 

period  would  be  -^=  ,  the  period  of  the  fundamental  tone  being  =  ; 

therefore  the  wave  length  is  twice  the  length  of  the  rod.  For  the 
general  case  in  which  there  is  a  node  at  the  middle  of  the  rod, 

cos  -j-,  vanishes  for  all  values  of  i  when  x  =  ~-    Then  A^  BI, 

I  A 


138  ELEMENTS    OF    WAVE    MOTION. 

must  vanish  for  all  even  values  of  i.  The  gravest  tone  is  then  the 
fundamental  tone  of  the  rod,  and  the  higher  tones  of  even  orders 
disappear.  The  first  tipper  tone  will  be  a  twelfth  above  the  funda- 
mental. In  this  case,  the  middle  section  might  become  absolutely 
fixed,  and  either  half  be  taken  away  without  disturbing  the  motion, 
so  as  to  leave  a  rod  of  half  the  length,  with  one  free  and  one  fixed 
end.  Therefore  the  fundamental  tone  of  a  rod  with  one  end  fixed 
is  the  same  as  that  of  a  free  rod  of  twice  the  length.  The  wave 
length  is  then  four  times  the  length  of  the  rod,  and  the  even  orders 
of  the  harmonics  are  wanting. 

225.  The  vibrations  of  air  columns  are  theoretically  the  same  as 
that  of  a  free  rod  or  one  fixed  at  an  end,  and  the  same  conclusions, 
modified  by  the  elasticity  of  the  air  and  its  velocity  of  wave  propa- 
gation, will  theoretically  apply.     We  will,  however,  determine  the 
positions  of  the  nodes  and  ventral  segments  of  vibrating  air  columns 
in  a  simpler  manner. 

226.  Vibrations  of  Air  Columns.    We  will  first  sup- 
pose a  single  sonorous  pulse  moving  in  an  air  column,  and  consider, 
1°,  the  column  closed  at  one  end  and  open  at  the  other.     Each 
stratum  passes  through  all  changes  of  density  during  the  periodic 
time  r,  while  the  pulse  moves  a  distance  A  ;  the  air  particles  de- 
scribe longitudinal  vibrations,  whose  amplitudes  depend  on  the 
intensity  of  the  sound.     When  the  condensation,  which  we  suppose 
is  in  advance,  reaches  the  closed  end,  the  air  stratum  at  that  place, 
not  having  freedom  of  motion,  undergoes  changes  of  density  alone. 
These  changes  are  each  immediately  reflected  in  succession,  and  the 
condensation  moves  from  the  closed  end  with  the  same  velocity  with 
which  it  would  have  proceeded  beyond  had  there  been  no  obstruc- 
tion to  its  progress.     Hence  we  see  that  at  the  instant  the  rarefac- 
tion first  reaches  the  closed  end  the  reflected  condensation  affects 
the  same  strata  as  the  incident  rarefaction,  and  disregarding  the 
loss  due  to  incidence,  the  air  strata  will,  at  this  instant,  in  the 

length  -  from  the  closed  end,  Have  their  normal  density  through- 

/c 

out.  The  velocities  of  the  air  particles,  at  the  same  instant,  will 
likewise  be  that  compounded  algebraically  of  those  belonging  to  the 
reflected  condensation  and  incident  rarefaction.  Now  when  a 
sonorous  body  is  vibrating,  the  sound  undulations  follow  each  other 


RELATING    TO    SOUND    AND    LIGHT. 


139 


periodically,  and  therefore  the  reflected  and  incident  pulses  will  be 
distributed  throughout  the  column.  The  densities  and  motions  of 
the  strata  will  therefore  result  from  the  combination  of  the  same 
elements  in  the  incident  and  reflected  pulses. 

227.  Let  the  curve  rm"mA  represent  the  direct  wave  at  any 
instant,  and  its  ordinates  the  corresponding  compressions  and  dila- 
tations of  the  air  on  the  line  rib'  due  to  this  wave;  the  curve 
m"m't)A.  and  its  ordinates  will,  in  like  manner,  represent  the  re- 
flected wave  from  the  stopped  end  AA'. 


Figure  3O. 


We  see  that  at  points  such  as  v,  v',  etc.,  at  JA,  JA,  etc.,  from 
AA',  the  condensations  or  dilatations  due  to  the  direct  wave  will 
always  be  contrary  and  equal  to  the  dilatations  or  condensations  due 
to  the  reflected  wave;  hence,  at  these  points,  the  normal  density  of 
the  air  will  ever  exist.  But  at  points  such  as  n,  ri,  etc.,  at  distances 
of  -|-/l,  A,  f  A,  etc.,  from  AA',  the  condensations  or  dilatations  of  each 
are  of  equal  value,  and  of  the  same  kind,  and  exist  simultaneously ; 
therefore  the  resultant  condensation  or  dilatation  is  double  that  due 
to  either.  At  these  points  then  the  air  undergoes  all  variations  of 
•density  during  the  period  r.  The  density  at  all  points  from  n  to  v 
and  to  v',  undergoes  decreasing  variations  from  the  maximum  at  n 
to  zero  at  v  and  v'. 

228.  With  regard  to  the  velocities  of  the  air  particles  at  differ- 
ent distances  from  AA',  since  the  motions  of  the  particles  change 
•direction  abruptly  at  reflection,  the  ordinates  of  the  curve  A'bmo 
will  represent  the  velocities  due  to  the  reflected  wave,  and  those  of 
Amm"r  may  now  represent  those  of  the  direct  wave.  Then  at  v,  v'9 

etc.,  the  velocities  are  zero  only  at  instants  separated  by  -,  and  at 

<o 

all  other  times  have  values  that  vary  from  zero  to  that  represented 
by  double  the  maximum  ordinate ;  at  n,  ri,  etc.,  the  velocities  are 


140  ELEMENTS    OF    WAVE    MOTION. 

always  zero,  and  therefore  the  air  at  these  points  is  quiescent,  while 
undergoing  changes  of  density.  At  intermediate  points,  both 
changes  in  velocities  and  density  occur. 

Hence,  we  conclude  that  nodes  will  be  developed  in  a  column  of 
air  closed  at  one  end,  when  it  is  traversed  by  a  sonorous  wave,  at 

distances  from  the  stopped  end  of  0,  -£ ,  — ,  — ,  etc. 

The  vibrating  parts  between  the  nodes  are  called  ventral  seg- 
ments, and  their  middle  points  are  at  distances  of  ^,  — ,  — ,  etc.,, 
from  the  stopped  end. 

229,  2°.  Open  Air  Columns.  Let  the  two  tubes  AM 
and  MB,  of  unequal  diameter,  be  united  at  M,  and  admit  that  there 
is  no  abrupt  change  of  density  of  the  air  at  M.  The  consequence 
of  a  contrary  supposition  is  that  the  opposite  sides  of  the  infinitely 
thin  stratum  M  would  be  subjected  to  unequal  pressures,  whose 
finite  difference  wonld  generate  in  M  an  infinite  velocity  in  a  finite 
time.  Hence,  the  density  has  the  property  of  continuity  in  its 
variation  throughout  AB.  It  is  not  essential  that  the  variation  of 
the  velocities  of  the  particles  of  air  should  be  continuous,  nor  is  it 
incompatible  with  this  condition. 

Let  s  and  sf  be  the  areas  of  sections  M      B 

in  AM  and  MB,  indefinitely  near  M;     — 
v  and  v'  the  velocities  of  the  air  parti-     


cles  in  s  and  s',  at  the  time  t ;  then 

vsdt  and  v's' dt  will  be  the  volume  of  Figure  31, 

air  passing    s  and   s'  during  dt,  and 

(vs  —  v's')  dt  will  be  the  increment  of  the  quantity  of  air  in  the 

volume  ss'  in  the  time  dt,  which  will  be  proportional  to  the  increase 

of  density  in  ss'.     But  in  order  that  the  increment  of  density  may 

be  compatible  with  the  supposed  continuity  of  the  pressure,  it  is 

evident  that  (vs  —  v's')  dt  must  be  an  infinitesimal  of  the  second 

order,  and  equal  to  zero  when  s  and  s'  are  coincident.     Hence,  at 

the  limit  we  have 

vs  =  v'sr  ; 

therefore,  there  will  be  a  wave  propagated  in  MB,  whose  intensity, 
determined  by  the  value  of  v',  will  become  more  and  more  inappre- 
ciable as  s'  becomes  greater  and  greater  than  s.  Let  MB  be  in* 


RELATING    TO    SOUND    AND    LIGHT.  141 

creased  indefinitely  in  area,  as  when  the  tube  AM  opens  into  the 
external  air,  then  v'  becomes  very  small,  and  the  transmitted  wave 
becomes  negligible,  as  is  the  case  in  open  pipes.  There  will  then 
be  a  reflected  wave  in  AM,  composed  of  a  rarefaction  followed  by 
a  condensation,  when  the  direct  wave  is  a  condensation  followed  by 
a  rarefaction.  The  velocities  of  the  air  particles  will  then  be  theo- 
retically equal  in  value,  and  the  same  in  direction  in  the  two  waves. 
The  curves  of  Fig.  30  will  illustrate  the  case  of  open  pipes,  if  A'bof 
represent  the  densities  and  Abm"f  the  velocities  of  the  air  particles 
in  the  reflected  Avave.  The  nodes  and  middle  points  of  the  ventral 
segments  will  then  be  at  distances  of 

A      3A      5A      7A 

4>    T>    T'    T'     ltc" 

2A      4A      6A 
and  0,     --,     — ,     -j-,    etc., 

from  M,  the  open  end  of  the  tube,  respectively. 

230.  These  laws,  which  determine  the  positions  of  the  nodes 
and  ventral  segments  of  vibrating  air  columns,  are  known  as  Ber- 
nouilli's  laws.  F/om  them  we  see  that  the  harmonics  of  open  pipes 
are  in  the  order  of  the  natural  numbers,  and  that  those  of  closed 
pipes  are  as  the  odd  numbers.  Thus,  the  open  pipe  can  give,  by 
an  increased  pressure,  the  octave,  the  twelfth,  the  fifteenth,  etc., 
while  the  closed  pipe  gives  the  twelfth,  the  seventeenth,  etc.  Ex- 
periments with  organ-pipes  verify  the  laws  of  Bernoulli!  only  approx- 
imately ;  that  is,  that  the  nodes  are  not  exactly  at  the  positions 
defined  above,  nor  are  the  nodes  exactly  places  of  rest.  Organ- 
pipes  are  usually  made  to  speak  by  forcing  a  current  of  air  through 
a  narrow  slit,  and  causing  it  to  impinge  against  a  thin  lip.  Of  the 
many  vibratory  motions  produced  in  this  manner,  there  is  always 
one  whose  periodicity  is  such  that,  by  the  resonance  of  the  pipe,  its 
intensity  will  be  raised  to  such  a  degree  as  to  produce  a  marked  and 
determinate  musical  sound,  called  the  fundamental  tone  of  the 
pipe.  Other  vibratory  motions,  which  undoubtedly  exist,  are  either 
destroyed  by  the  interference  of  the  reflected  waves,  or  have  so  feeble 
an  intensity  as  to  be  negligible.  The  wave  length  of  the  funda- 
mental tone  is,  as  we  have  seen  above,  double  the  length  of  the 
open  pipe,  or  four  times  the  length  of  the  closed  pipe,  approxi- 


14:2  ELEMENTS    OF    WAVE    MOTION. 

raately.  The  discrepancy  between  experiment  and  theory  arises 
from  the  fact  that  the  hypothesis  is  not  in  accord  with  what  actu- 
ally occurs  in  the  pipe.  Without  considering  these  minutely,  it  is 
sufficient  to  note  the  perturbations  at  the  embouchure  by  the  air 
current,  the  modifications  in  the  pipes  by  the  moving  air,  and  the 
induced  vibrations  of  the  material  of  the  pipe  at  the  sides  and 
closed  end,  to  account  for  the  greater  discrepancies. 

231.  Relative    Velocities    of  Sound   in    Different 
Material.    Since  in  any  medium,  we  have  A  —  Vr  =  V-,  in 

which  n  is  the  vibrational  number  for  a  note  of  definite  pitch,  A  the 
corresponding  wave  length  in  the  same  medium,  and  V  the  velocity 
of  sound,  it  is  readily  seen  that  if  free  rods  of  different  material  be 
taken,  of  such  lengths  as  to  give  the  same  note  when  put  into 
longitudinal  vibration,  we  will  have 

A  =  V-,        A'  =  V-,        A"  =  V"-,    etc.; 
n'  n  n 

whence 

A:  A':  A"  ::  V:  V  :  V". 

A    A'    A" 
But  as  ^,  —  ,  —  ,  etc.,  are  the  lengths  of  free  rods  that  give  the 

<i       6  A 

fundamental  tone,  we  see  that  such  lengths  are  directly  propor- 
tional to  the  velocities  of  sound  in  the  several  media,  when  their 
lengths  are  great  compared  to  their  cross-sections.  Knowing  then 
the  velocity  of  sound  in  any  material,  we  can  by  experiment  find 
that  in  others  by  this  method.  Then  having  the  velocities,  we  can 
by  substitution  in  the  formula 


find  the  value  for  the  longitudinal  modulus  E. 

232,   Applying  the  same  principle  to  any  gas  and  comparing  the 
velocity  in  it  with  that  m  air,  by  the  formula 


RELATING    TO    SOUND    AND    LIGHT. 


143 


the  values  of  y,  or  the  ratio  of  its  specific  heats,  can  be  readily 
obtained.     By  this  means  Dulong  found  the  following  results  : 


DENSITY. 

VELOCITY. 

c 

Y~^' 

Air    

1. 

333. 

1.421 

Oxygen  . 

1.1026 

317.7 

1.415 

Hydrogen  ...         ... 

0.0688 

1269.5 

1.407 

Carbon  Dioxide  ... 
Carbon  Monoxide,  .... 

1.524 
0.974 

261.6 
337.4 

1.338 
1.427 

Under  the  assumption  that  the  gas  is  perfect,  simple,  and  far 
from  its  point  of  liquefaction,  y  is  assumed  to  have  the  constant 
value  of  1.41.  The  above  results  show  that  this  value  should  be 
considered  as  the  limit  to  which  y  approximates  and  only  reaches 
under  the  particular  suppositions  made. 

233.   Transversal  Vibration  of  Elastic  Hods.    An 

elastic  rod  is  a  rigid  body  whose  cross-section,  considered  uniform 
throughout,  is  taken  as  very  small  compared  with  its  length.     The 
rod  or  bar  may  be  arranged  in  six  different  ways,  depending  on  the 
method  by  which  its  ends  are  sustained,  viz. : 
1°.  The  rod  may  be  free  at  both  ends. 
2°.  It  may  be  firmly  fixed  at  both  ends. 

It  may  be  fixed  at  one  end  and  free  at  the  other. 
It  may  be  supported  at  one  end  and  free  at  the  other. 
It  may  be  fixed  at  one  end  and  supported  at  the  other. 
It  may  be  supported  at  both  ends. 
It  may  yield  its  fundamental  tone  by  vibrating  as  a  whole,  or 
give  tones  of  higher  pitch  by  dividing  itself  into  vibrating  parts 
separated  by  nodes.    The  formula 


(279) 


gives  the  number  of  vibrations  in  all  cases,  as  has  been  verified  by 
experiment.  In  this  formula,  N -is  the  number  of  vibrations  per 
second ;  n  a  constant  depending  on  the  manner  in  which  the  rod  is 
arranged  at  the  ends  and  on  the  number  of  nodes  formed  ;  t  is  the 


3°. 

4°. 

5°. 
6°. 


144  ELEMENTS    OF    WAVE    MOTION. 

thickness,  measured  in  the  plane  of  vibration  ;  I  is  the  length,  E 
the  rigidity,  and  D  the  density  of  the  rod. 

234.  This  formula  shows  that  the  vibrational  number  is  inde- 
pendent of  the  width,  provided  it  be  small  as  at  first  supposed  ; 
that  it  is  directly  proportional  to  the  thickness,  inversely  as  the 
square  of  the  length,  and  directly  as  the  square  root  of  the  rigidity 
divided  by  the  density. 

1°.  The  rod  is  free  at  both  ends.  Lissajous  has  determined 
by  careful  experiments  that  the  following  formulae  apply,  viz.  : 


in  which  I  is  the  length,  n  the  number  of  nodes  formed,  d  the  dis- 
tance between  two  consecutive  nodes,  6-  the  distance  from  the  free 
ends  to  the  nearest  nodes,  and  s'  the  distance  from  the  free  ends  to 
the  second  nodes.  Hence  from  these  formulae,  we  see  that  the 
intermediate  nodes  are  equidistant;  that  the  distance  from  the 
extreme  nodes  to  the  next  adjacent  is  nearly  0.92  of  the  distance 
between  two  consecutive  intermediate  nodes  ;  that  sis'  ::  0.2643  :  1, 
and  s:d  ::  0.33  :  1.  Experiment  confirms  these  results  whatever  be 
the  number  of  the  nodes.  The  positions  of  the  nodes  are  made  visi- 
ble by  sprinkling  sand  on  the  bar,  and  noticing  the  lines  on  which 
it  accumulates  when  the  bar  or  rod  is  put  in  vibration. 

2°.  Both  ends  are  fixed.  When  the  ends  are  so  fixed  as  not  to 
modify  its  elasticity  at  these  points,  it  can  vibrate  freely,  and  the 
nodes  are  found  to  be  located  at  the  same  places  as  in  a  free  rod  of 
the  same  length,  except  that  the  extreme  nodes  are  at  the  fixed 
ends.  The  first  two  of  formulae  (280)  are  then  applicable  to  this 
case. 

3°.  The  rod  is  fixed  at  one  end  and  free  at  the  other.  There 
will  then  be  0,  1,  2,  3,  ....  nodes  depending  upon  the  manner  by 
which  it  is  put  in  vibration.  If  the  fixed  end  be  regarded  as  a 
node,  the  first  of  the  above  formulas  is  applicable,  and  the  other 
two  apply  to  the  free  end  only.  Therefore  these  first  three  cases 
are  all  reducible  with  the  modifications  mentioned  to  that  of  a  free 
rod  at  both  ends. 

4°  and  5°.  In  these  cases  the  supported  end  may  be  considered 
as  an  intermediate  node,  and  we  can  consider  the  rod  as  half  of  a 
rod  of  double  the  length,  free  or  fixed  at  both  ends  in  which  the 


RELATING    TO    SOUND    AND    LIGHT.  145 

number  of  nodes  is  2n  —  l.     Replacing  I  by  21  and  n  by  2n  —•  1, 
we  then  have 

7  41  ,51  1.3216  / 

J=='  =  : 


of  which  the  last  two  apply  only  to  the  case  where  one  of  the  ends 
is  free. 

6°.  If  the  supported  ends  be  regarded  as  intermediate  nodes 
we  have 


235,  Harmonic  Vibrations  of  Elastic  Hods.  When 
the  vibrating  parts  are  known,  the  harmonies  of  the  rod  are  easily 
determined,  and  considering  the  fixed  extremities  as  nodes,  the 
formulae  of  Lissajous  above  given  become  general  for  the  six  cases. 
In  the  first  three  cases  the  sounds  resulting  are  the  same  for  the 
same  number  of  nodes,  whatever  be  the  condition  of  the  extremity, 
whether  fixed  or  free.  The  numbers  of  vibrations  are  as  32,  52,  72, 
----  (2n  —  I)2,  when  there  are  2,  3,  4,  ----  n  nodes.  In  the  4°  and 
5°  cases  where  one  of  the  extremities  is  supported,  the  vibrational 
numbers  are  as  52,  92,  132,  ----  (4ra  —  3)2;  and  in  the  6°  case  the 
numbers  are  I2,  22,  32,  ----  (n  —  I)2,  n  being  the  number  of  nodes. 
Comparing  in  all  these  cases  the  vibrational  numbers  for  two 
nodes,  we  have 

0     25  9     25 

9:T:4>        °r        4:16:1' 

and  therefore  generally 


4  16 

For  d  we  have 

21  4? 


2n  —  l'        4^  —  3'        n  —  l 

Substituting  the  value  of  n  taken  from  the  latter  in  the  former, 
we  have 


~ 


(383) 


If  d  =  ?,  which  corresponds  to  a  rod  supported  at  both  ends  and 
yielding  its  fundamental  sound,  we  have  N  =  1.    We  therefore  con- 
10 


146 


ELEMENTS    OF    WAVE    MOTION. 


elude  that  when  a  rod  gives  a  harmonic,,  the  parts  comprised  between 
the  nodes  vibrate  as  rods  whose  extremities  are  supported  and  whose 
length  is  the  distance  between  the  nodes,  and  that  the  vibrational 
number  is  inversely  as  the  square  of  this  length.  This  conclusion 
is  inapplicable  to  the  first  nodes,  because  they  are  more  or  less 
influenced  by  the  extremities. 

236.  Tuning  Forks.  A  tuning  fork  may  be  regarded  as  a 
rod  or  bar  free  at  both  ends.  Experiment  shows  that  in  proportion 
as  a  bar  free  at  both  ends  is  bent  or  curved  the  extreme  nodes 
approach  each  other.  Thus,  in  the 
figure  the  bar  ab  if  supported  at  the 
points  1,  2,  one  fourth  the  length 
of  the  bar  from  the  extremes,  will 
when  vibrated  transversely  develop 
nodes  at  these  points.  In  the  forms 
a'V,  a"b",  a"'V",  the  length  remain- 
ing unchanged,  the  nodes  approach 
each  other  as  indicated  in  the  figure. 
The  laws  which  govern  the  vibration 
of  a  fork  whose  section  is  rectangular  have  been  experimentally 
found  to  be  ;  1°,  that  the  vibrational  number  is  independent  of  the 
width  ;  2°,  proportional  to  the  thickness  ;  3°,  inversely  proportional 
to  the  square  of  the  length  increased  slightly.  The  length  is  taken 
as  equal  to  the  projection  of  the  prongs  on  the  medial  line  of  the 
fork.  For  a  fork  of  rectangular  cross-section  we  have  from  the 
experiments  of  Mercadier. 

(284) 


Figure  32, 


in  which  N  is  the  vibrational  number,  t  the 
thickness,  and  I  the  length  ;  «  is  a  constant 
which  for  steel  is  found  to  be  818270.  When 
the  fork  yields  its  fundamental  note  its  method 
of  division  is  shown  in  the  figure.  The  over- 
tones of  a  fork  correspond  to  vibrational 
numbers  which  are  to  each  other,  beginning 
with  the  first  ;  as,  32  :  52  :  72  :  etc.  The  vibra- 

25 
ticnal  number  of  .the  first  overtone  is  about  -- 


T 


¥ 


Figure  33. 


RELATING    TO    SOUND    AND    LIGHT.  147 

that  of  the  fundamental.  Helmholtz  found  by  experimenting  on 
many  forks,  that  it  varied  from  5.8  to  6.6  that  of  the  funda- 
mental. These  overtones  are  so  high,  that  they  are  generally  of 
short  duration,  and  they  are  also  inharmonic  with  the  prime. 
Tuning  forks  are  generally  mounted  on  their  resonant  boxes,  by 
which  arrangement  the  prime  tone  of  the  fork  is  greatly  rein- 
forced to  the  disadvantage  of  the  overtones.  The  duration  of 
the  vibration  of  a  fork  although  theoretically  constant,  is  found 
to  increase  slightly  with  an  increase  of  amplitude  and  tempera- 
ture, thus  slightly  lowering  the  pitch.  This  is,  however,  not 
appreciable  to  the  hearing,  but  can  be  detected  by  any  of  the  graph- 
ical methods  for  determining  the  number  of  vibrations  in  a  given 
period.  It  is  a  matter  of  importance  in  determining  the  initial 
velocity  of  projectiles,  by  means  of  the  Schultz  chronoscope  or  other 
devices,  where  the  vibrations  of  a  tuning  fork  enter  into  the  calcula- 
tion, to  limit  the  amplitude  and  to  take  note  of  the  temperature,  in 
order  to  obtain  uniform  and  reliable  results.  When  the  amplitude 
does  not  surpass  3  or  4  mm.  and  the  temperature  varies  but  little 
beyond  the  ordinary  atmospheric  temperature,  the  vibrational 
number  may  be  taken  as  constant  within  .0001  of  its  value. 

237.  Vibration  of  Plates.  Plates  are  rigid  bodies,  gener- 
ally of  metal  or  glass,  whose  length  and  breadth  are  very  great 
compared  with  their  thickness.  To  put  them  in  vibration,  one  or 
more  points  are  fixed  and  a  violin  bow  is  drawn  across  an  edge. 
The  circumstances  of  vibration  are  exhibited  by  sprinkling  fine 
sand  over  the  surface  and  examining  the  nodal  lines  formed  by  the 
sand  which  seeks  that  part  of  the  plate  which  is  at  rest.  The  parts 
of  the  plate  separated  by  a  nodal  line,  evidently  vibrate  in  opposite 
directions,  and  therefore  for  permanent  figures  the  number  of 
vibrating  parts  must  be  even.  When  the  plate  yields  its  funda- 
mental tone  the  resulting  figure  is  the  simplest  that  can  be  formed, 
and  as  the  plate  separates  into  a  greater  number  of  vibrating  parts, 
the  figures  become  more  complex.  Chladni  has  given  to  these 
figures  the  name  of  Acoustic  figures.  As  yet,  from  the  inherent 
difficulties  of  the  problem,  the  mathematical  laws  have  not  been 
deduced,  but  experiment  has  assigned  the  following  as  the  laws  of 
vibrating  plates,  viz. ;  1°,  the  vibrational  numbers  of  plates  of  the 
same  form  and  of  the  same  material  are  inversely  as  the  squares  of 


148 


ELEMENTS    OF    WAVE    MOTION. 


the  homologous  dimensions  ;  2°,  and  are  proportional  to  the  thick- 
ness.    Hence  we  have 

t_   V_ 

P ''  I'*' 


n  :  n 


If  a  rectangular  plate  be  so  constructed  that  a  system  of  nodal 
lines  parallel  to  the  length  be  formed  by  a  sound,  which  gives 
another  system  of  nodal  lines  parallel  to  the  breadth,  when  it  is 
vibrated  in  these  two  ways,  then  if  at  any  of  the  middle  points 
of  the  ventral  segments  it  be  vibrated  so  as  to  produce  the  same 
sound,  these  two  systems  will  simultaneously  exist  and  the  acoustic 
figure  will  result  from  the  combination  of  these  two  systems.  The 
figure  illustrates  five  such  plates  where  the  numbers  of  the  nodal 
lines  are  in  the  ratios  of  2 : 3,  2 : 4,  3:4,  3:5,  4:5.  Other  combi- 
nations illustrative  of  the  vibrations  of  plates  are  reserved  for  the 
lectures. 


S:6 


4:5 


Figure  34i 


238.  Vibration  of  Membranes.  When  a  stretched  mem- 
brane is  near  a  sounding  body,  the  air  transmits  to  it  the  vibratory 
motion.  It  can  respond,  however,  only  to  certain  sounds  depend- 
ing on  its  tension,  and  thus  enter  into  synchronous  vibration. 
This  fact  is  made  evident  by  the  acoustic  pendulum,  or  by  the 
nodal  lines  formed  by  sand  sprinkled  upon  it,  as  in  the  case  of  the 


RELATING    TO    SOUND    AND    LIGHT.  149 

vibration  of  plates.  The  frames  upon  which  the  membranes  are 
stretched  are  generally  square  or  circular.  Experiment  has  con- 
firmed the  following  deductions  of  Poisson  and  Lame,  with  respect 
to  the  vibrations  of  square  membranes,  viz.  : 

1°.  Membranes  respond  only  to  certain  sounds,  separated  by 
determinate  intervals. 

2°.  To  each  sound  a  system  of  nodal  lines  corresponds,  parallel 
to  the  sides  of  the  membrane,  and  whose  numbers  are  represented 
by  n  and  ri. 

3°.  The  nodal  lines  which  correspond  to  the  same  sound  form  a 
system  of  figures,  such  that  we  can  pass  from  one  to  the  other  by 
continuous  changes  in  varying  the  mode  of  disturbance,  without 
changing  the  sound ;  but  we  can  never  pass  in  a  continuous  man- 
ner from  the  lines  of  one  sound  to  those  of  another. 

Circular  membranes  can  only  give  nodal  lines  along  the, diam- 
eters or  circumferences,  either  separate  or  combined,  depending  on 
the  method  of  vibration  and  on  the  point  or  points  of  enforced  rest. 

239.  Because  of  the  limited  time  allotted  to  this  part  of  the 
course,  many  subjects  of  importance  are  necessarily  omitted  in  the 
text.     Among  these  are, 

1°.  The  theory  of  beats,  and  resultant  sounds. 

2°.  The  phenomena  of  interference,  whose  consequences,  how- 
ever, are  readily  derived  from  the  discussion  in  Arts.  65-68. 

3°.  The  graphical  and  optical  methods  of  the  study  of  sonorous 
vibrations,  and  that  by  sensitive  and  manometric  flames. 

4°.  The  phenomena  of  vibrations  of  air  columns  in  organ-pipes, 
of  elastic  rods,  of  plates  and  membranes,  with  the  applications  of 
the  latter  in  the  phonograph,  phonautograph,  and  telephone. 

By  means,  however,  of  a  very  complete  acoustical  apparatus, 
mainly  from  the  workshop  of  Koenig,  the  celebrated  physicist  of 
Paris,  the  omitted  parts,  as  well  as  those  treated  of  above,  are  illus- 
trated in  the  lectures,  which  largely  supplement  and  complete  the 
study  of  the  text. 

240.  The  nature  and  essential  principles  of  undulatcry  motion, 
as  illustrated  by  sonorous  vibrations,  have  received  sufficient  atten- 
tion to  enable  the  student  to  prosecute  understandingly  the  study 
of  similar  principles  connected  with  light  in  the  analogous  subject 
of  optics. 


PART    III. 
OPTICS. 


241.  Light  is  the  agent  by  which  the  existence  of  bodies  is 
made  known  to  us  through  the  sense  of  sight. 

That  branch  of  physical  science  which  treats  of  the  properties 
of  light  and  the  laws  of  its  transmission  is  called  Optics. 

242.  It  is  divided  into  two  parts  : 

1°.  Geometrical  Optics,  which  embraces  all  the  phenom- 
ena relating  to  the  propagation  of  rays,  based  on  certain  experi- 
mental laws,  and  which  is  entirely  independent  of  any  theory  as  to 
the  nature  of  the  luminous  agent. 

Experiments  in  Geometrical  Optics,  however  carefully  made, 
can  never  accurately  prove  the  laws  of  light  propagation,  but  serve 
merely  to  establish  a  certain  degree  of  probability  of  their  truth, 
and  which,  when  applied  to  other  phenomena  of  the  same  nature, 
strengthen  this  probability  in  proportion  as  the  application  is  more 
extended. 

2°.  Physical  Optics9  which  is  based  on  the  theory  of  un- 
dulations, and  seeks  to  explain  by  this  theory  the  nature  of  light, 
and  of  all  the  phenomena  arising  from  the  action  of  rays  on  each 
other. 

243.  That  light  is  not  a  material  substance,  but  is  merely  a 
process  going  on  in  some  medium,  is  proved  by  the  phenomena  of 
interference,  in  which  results  of  various  magnitudes  occur,  from 
less  to  greater,  or  the  reverse,  depending  upon  the  manner  in  which 
the  interference  takes  place,  even  when  the  combining  magnitudes 
are  themselves  constant  in  value. 

244.  The  undulatory  theory  asserts  that  light  is  due  to  the 
transmission  of  energy  from  luminous  bodies  to  the  finely-divided 
parts  of  the  optic  nerve,  spread  over  the  interior  concave  surface  of 


RELATING    TO    SOUND    AND    LIGHT.  151 

the  eye.  This  energy  is  conveyed  by  the  optic  nerve  to  the  brain, 
and  there  transformed  into  the  sensation  of  sight. 

The  transmission  of  the  energy  is  accomplished  by  undulatory 
motion  in  a  medium  called  the  luminiferous  ether.  There  is  no 
direct  proof  of  the  actual  existence  of  the  ether,  and  its  assumption 
can  only  be  regarded  as  an  extremely  probable  hypothesis,  supported 
by  nearly  all  the  known  phenomena  of  light,  and  directly  contra- 
dicted by  none. 

Within  the  present  century,  its  reality  has  been  almost  uni- 
versally accepted,  and  as  a  consequence  the  undulatory  theory  has 
entirely  supplanted  the  rival  hypothesis  of  the  materiality  of  light 
.  molecules,  known  as  the  emission  theory,  which  had,  however,  held 
its  ground  for  many  years. 

245.  The  accepted  properties  of  the  luminiferous  ether  have 
resulted  from  theoretical  considerations,  modified  from  time  to  time 
by  deductions  from  experimental  observations,  and  while  there  are 
several  imperfections  yet  to  be  removed,  nevertheless  the   strong 
array  of  unquestioned  facts,  both  observed  and  predicted,  has  estab- 
lished these  properties  as  a  satisfactory  foundation  upon  which 
modern  physical  optics  is  now  constructed. 

The  luminiferous  ether  is  considered  to  be  a  material  substance 
of  a  more  rare  and  subtile  nature  than  the  ordinary  matter  affecting 
the  senses,  and  to  exist  not  only  within  these  bodies,  but  through- 
out space.  It  has  great  elasticity,  and  is  capable  therefore  of  trans- 
mitting its  particular  energy  over  vast  distances,  with  great  velocity 
and  with  inappreciable  loss.  That  this  energy  is  not  transmitted 
instantaneously  has  been  proved  by  direct  experiment,  and  con- 
cluded from  several  astronomical  observations. 

246.  That  light  is  propagated  in  right  lines  from  the  source  is 
a  fact  of  observation  and  experiment.     This  statement,  however, 
while  absolutely  true,  is  subject  to  modification  when  taken  in  the 
ordinary  sense  of  the  language.     Thus,  we  have  seen  that  while 
sound  is  propagated  in  right  lines  from  its  source,  it  is  capable  of 
spreading  around  an  obstacle,  so  that  sound  can  be  heard  out  of  the 
direct  line  of  the  source  ;  so,  in  a  less  degree,  we  can  see  around  an 
obstacle,  as  will  be  shown  in  the  discussion  of  the  diffraction  of 
light. 

The  acoustic  shadow,  however,  is  as  much  less  marked  than  the 
optical  shadow  as  the  wave  lengths  of  sound  are  greater  than  the 


152  ELEMENTS    OF    WAVE    MOTION. 

wave  lengths  of  light.     But  for  the  explanation  of  the  principles  of 
geometrical  optics  it  is  unnecessary  to  consider  this  refinement. 

247.  Bodies  are  called  self-luminous  when  they  are  themselves 
the  sources  of  light,  and  rays  proceed  directly  from  them.     They 
are  visible  because  of  their  emanating  rays.     Other  bodies  are  called 
non-luminous,   and  become  visible   because  rays  from  luminous 
bodies  are  reflected  from  their  surfaces. 

A  luminous  point,  or  origin  of  light,  is  a  very  small  portion  cf  a 
luminous  surface.  When  light  emanates  from  a  luminous  point, 
we  consider  it  made  up  of  rays  of  light,  each  of  which  is  the  small- 
est portion  of  light  which  can  be  transmitted.  The  ray  is  the  right 
line  along  which  the  undulation  is  propagated,  and  is  practically  a 
mere  conception,  indicating  direction. 

A  collection  of  parallel,  diverging,  or  converging  luminous  rays 
is  called  a  beam  of  light,  and  sometimes  &  pencil  of  light,  the  latter 
name  being  generally  applied  to  the  last  two  cases. 

The  axis  of  a  beam  is  the  geometrical  axis  of  the  cylinder  or 
cone  of  rays  ;  when  the  axis  is  normal  to  the  deviating  surface,  the 
beam  is  direct,  and  when  inclined  to  it,  oblique. 

248,  When  a  beam  of  light  is  incident  upon  any  surface,  it  is 
generally  separated  into  three  portions,  viz.,  a  part  is  scattered  or 
diffused  over  the  surface,  by  which  the  surface  becomes  visible,  a 
second  part  is  reflected,  and  the  remainder  is  refracted. 

The  proportion  of  the  several  parts  depends  on  the  polish  of  the 
surface,  the  angle  of  incidence,  and  the  nature  of  the  medium.  A 
perfectly  polished  surface  would  be  invisible,  and  the  incident  beam 
would  be  separated  into  a  reflected  and  refracted  beam  alone ;  of 
course,  such  a  polish  is  not  practicable.  Light  regularly  reflected 
has  its  intensity  increased  with  the  degree  of  polish,  while  the  in- 
tensity of  irregularly  reflected  light  is  similarly  diminished.  The 
intensity  of  regularly  reflected  light  from  the  surface  of  water  is,  at 
the  incidences  of 

0°,  40°,  60°,          80°,         89i°, 

about  1.8$,        2.2$,         6.5$,        33$,         72$. 

At  normal  incidence,  water,  glass,  and  mercury  reflect  1.8$, 
2.5$,  and  66f  $,  respectively.  The  differences  at  small  angles  of 
incidence  are  more  marked  than  at  greater  angles,  since  while  both 


RELATING    TO    SOUND    AND    LIJHT.  153 

water  and  mercury  reflect  the  same  at  89^°,  -the  former  reflects  but 
5^  as  much  as  the  latter  at  normal  incidence. 

249.  A  medium  is  any  substance  which  permits  the  passage  of 
light  through  it. 

Since  the  luminiferous  ether  is  supposed  to  pervade  all  matter, 
it  might  be  inferred  that  all  bodies  could  be  classed  under  the  head 
of  media  for  light.  Gold,  although  one  of  the  most  dense  of  sub- 
stances, does  permit  the  passage  of  light,  when  beaten  into  a  very 
thin  leaf ;  and  no  doubt  if  other  opaque  bodies  possessed  an  equal 
malleability,  the  same  property  would  belong  to  them. 

But  owing  to  internal  reflection  and  consequent  interference,  it 
is  assumed  that  an  inappreciable  quantity  of  light,  if  any,  passes 
through  very  small  thicknesses  of  opaque  bodies.  Glass,  air,  water, 
and  all  other  matter  which  permit  the  passage  of  light  freely,  are 
said  to  be  transparent.  Translucency  is  a  term  applied  to  such 
bodies  as  permit  the  passage  of  diffused  light;  thus,  ground  glass 
and  flint  are  translucent,  while  clear  glass  and  quartz  crystal  are 
transparent. 

250.  Since  light  is  assumed  to  result  from  undulatory  motion 
in  the  luminiferous  ether,  all  the  consequences  deduced  in  the  dis- 
cussion of  the  properties  of  this  kind  of  motion  in  Part  I  are  at 
once  applicable  to  the  phenomena  of  light. 

251.  Shadows  and  Shade.    From  each  luminous  point 
considered  as  an  origin  of  disturbance,  undulations  proceed  along 
right  lines  in  all  directions  from  this  origin.     Therefore,  whenever 
they  meet  an  opaque  body,  this  undulation  will  be  deviated  from 
its  original  direction,  and  the  effect  of  light  will  be  wanting  along 
this  direction  prolonged. 

The  absence  of  this  effect  is  called  the  shadow  of  the  point  of  the 
opaque  body. 

The  line  of  the  surface  of  the  opaque  body,  along  which  rays 
drawn  from  the  luminous  point  are  tangent,  is  called  the  line  of 
shade.  Since  each  point  of  the  luminous  surface  is  an  origin  of 
light,  we  see  that  in  all  actual  cases  the  shadow  of  an  opaque  body 
must  be  indistinct  near  its  boundary,  and  gradually  merge  into  the 
illuminated  surface  surrounding  the  shadow,  whenever  the  lumi- 
nous source  is  of  an  appreciable  area.  This  modified  portion  of  the 
shadow  is  due  to  the  overlapping  of  the  cones  of  rays  proceeding 


154 


ELEMENTS    OF    WAVE    MOTION. 


from  each  luminous  point,  and  is  called  the  penumbra.  It  is  lim- 
ited by  the  space  between  the  two  cones,  whose  elements  are  tan- 
gent to  the  luminous  surface  and  the  opaque  body,  one  having  its 
vertex  between  the  two,  and  the  other  its  vertex  on  the  further  side 
of  either  one  of  the  surfaces.  The  softness  of  shadows  in  general  is 
due  to  the  finite  extent  of  luminous  surfaces. 

252.   Every  point  of  the  luminous  source  emitting  rays  in  all 
directions,  each  will  carry  an  image  of  its  luminous  point. 


Thus,  if  a  lighted  candle  be  placed  in  front  of  a  small  aperture 
of  a  darkened  chamber,  the  aperture  will  permit  the  passage  of  a 
limited  number  of  the  rays  from  every  point  of  the  candle,  each  ray, 
however,  carrying  an  image  of  its  radiant.  The  image,  as  shown  in 
Figure  35,  will  be  inverted. 

If  another  aperture  be  made  near  the  first,  a  second  image  of 
the  candle  will  be  formed,  overlapping  the  first,  and,  while  the 
luminosity  will  be  increased,  the  image  will  lose  distinctness,  be- 
cause of  this  overlapping.  The  diffused  light  of  a  room  during  the 
day  is  due  to  the  overlapping  images  of  external  objects,  caused  by 
rays  proceeding  from  each  of  them,  thus  making  their  individual 
images  indistinct.  A  small  aperture  in  a  darkened  room  will  per- 
mit the  formation  of  an  inverted  image  of  the  external  scenery  upon 
a  screen  placed  within  the  room  near  the  aperture. 

253,  Photometry.  The  eye  possesses  the  property  of  dis- 
tinguishing color  and  intensity. 

In  determining  variations  of  intensity,  the  judgment  is  only 
approximate  when  the  colors  are  the  same,  and  the  difficulty  of  this 
appreciation  is  increased  when  the  colors  differ.  Equality  of  in- 


RELATING    TO    SOUND    AND    LIGHT.  155 

tensity  can  readily  be  determined  by  the  eye,  while  it  is  not  possible 
to  ascertain  the  numerical  ratio  of  different  intensities  by  direct 
observation. 

Photometry  has  for  its  object  the  measurement  and  comparison 
of  the  intensities  of  different  lights. 

254.  The  principle  of  all  photometric  methods  is  to  arrive  at 
this  comparison,  by  the  appreciation  of  the  equality  of  illumination 
of  two  near  surfaces,  physically  identical.  In  assuming  the  dis- 
tance of  the  luminous  source  from  the  illuminated  surface  to  be 
great  in  comparison  with  the  dimensions  of  the  surface,  and  remem- 
bering that  the  intensity  of  the  light  is  due  to  the  molecular  kinetic 
energy,  we  readily  see,  if  there  be  no  absorption  of  this  energy  dur- 
ing transmission  through  the  intervening  media, 

1°.  That  the  intensity  of  the  illumination  on  the  unit  area  of 
any  surface,  taken  normal  to  the  direction  of  propagation,  at  a 

distance  d  from  the  luminous  source,  varies  as  -^- 

a* 

2°.  That  if  /  represent  the  intensity  of  any  given  light,  and  if 
it  be.  supposed  to  illuminate  uniformly  any  area  A,  the  intensity  on 

a  unit  of  area  varies  as  —  • 
A 

3°.  That  the  quantity  of  light  emanating  from  any  luminous 
element,  and  hence  the  intensity  of  illumination  on  the  unit  area, 
is  proportional  to  the  cosine  of  the  angle  made  by  the  normal  to  the 
element  with  the  direction  considered,  and  hence  varies  as  the  co- 
sine of  the  inclination,  or  cos  i. 

4°.  That  if  the  area  on  which  the  light  falls  is  inclined  to  the 
direct  line  of  propagation,  the  illumination  on  the  unit  area  is  pro- 
portional to  the  cosine  of  the  angle  made  by  this  line  and  the  nor- 
mal to  the  surface,  or  to  cos  i'. 

5°.  That  the  illumination  on  the  unit  area  will  vary  with  the 
intrinsic  brightness  of  the  source.  The  intensity  of  the  illumina- 
tion on  the  unit  area,  parallel  to  the  source,  at  the  distance  unity, 
may  be  taken  as  the  measure  of  the  intrinsic  brightness. 

255.  Let  S  and  8'  be  the  projections  of  the  luminous  and  the 
illuminated  surfaces,  respectively,  on  a  plane  normal  to  the  direc- 
tion of  the  luminous  rays;  B  the  intrinsic  brightness  of  the 


156  ELEMENTS    OF    WAVE    MOTION. 

source  ;  d  the  distance  apart  of  the  two  surfaces,  and  /  the  intensity 
of  the  illumination ;  then,  from  the  above  principles,  we  have 

I=B8j^.  (385) 

Making  8'  =  1,  and  calling  /,  the  total  brilliancy  of  the  source 
at  the  distance  d,  we  have 

/,  =  *!•  (280) 

& 

-^  is  the  apparent  area  of  the  source  seen  from  the  illuminated 
surface,  and  making  this  equal  to  unity,  we  have 

/„  =  B.  (287) 

Therefore  the  intrinsic  brightness  of  the  source  is  the  total 
brilliancy  of  the  apparent  unit  of  area  of  the  luminous  surface  at 
the  distance  1. 

The  general  method  of  comparison  of  the  intrinsic  brightness  of 
two  sources  consists  in  permitting  the  rays  from  each  source  to  fall, 
nearly  normal,  upon  adjacent  portions  of  the  same  surface ;  then 
to  increase  the  distance  of  the  stronger  light,  until  the  eye  judges 
the  illumination  to  be  equal.  We  then  have 

BS  __  B,S,  (    v. 

~#-    ~d?> 

from  which,  by  substituting  the  known  values  of  d,  dn  S  and  $,, 
the  ratio  of  B,  to  B  can  be  determined. 

256.  The  apparent  intrinsic  brightness  of  an  object  is  equal  to 
the  quantity  of  light  received  from  it  by  the  eye,  divided  by  the 

area  of  the  picture  on  the  retina.     Therefore,  since  the  apparent 

<j 
illumination  of  the  object  is  B  -=- ,  and  the  area  of  the  retinal  pic- 

8 

ture  is  -„,  the  apparent  intrinsic  brightness  will  vary  with  the  real 

Cv 

intrinsic  brightness  B,  and  the  object  will  appear  equally  bright  at 
all  distances. 

This  result  is  deduced  under  the  supposition  that  no  light  from 
the  object  is  absorbed  by  the  medium  through  which  it  passes,  and 
is  therefore  only  an  approximation. 


RELATING    TO    SOUND    AND    LIGHT. 


157 


257.  Velocity  of  Light.  In  1675,  the  Danish  astronomer, 
Ucemer,  noticed  certain  discrepancies  with  regard  to  the  observed 
times  of  the  eclipses  of  Jupiter's  satellites,  which  he  correctly  at- 
tributed to  the  finite  velocity  of  light. 

To  show  this,  let  S  be  the  sun,  EE'  the  earth's  orbit,  JJ'  the 
orbit  of  Jupiter,  and  ss'  the  orbit  of  Jupiter's  inner  satellite. 


Figure  36. 


The  planets  and  satellites  shine  by  the  reflected  light  of  the  sun, 
and  therefore  cast  shadows,  whose  axes  are  on  the  right  lines  join- 
ing their  centres  with  the  centre  of  the  sun.  Because  of  the  posi- 
tion of  the  orbit  of  the  satellite  with  respect  to  the  plane  of  Jupiter's 
orbit,  the  satellite  enters  Jupiter's  shadow  at  every  revolution,  and 
is  eclipsed.  If  light  traversed  space  instantaneously,  its  entrance 
into  and  exit  out  of  the  shadow  might  be  noted  at  the  exact  in- 
stants at  which  these  phenomena  occurred,  independently  of  the 
relative  positions  of  the  earth  and  Jupiter. 

But  when  Jupiter  is  near  opposition,  as  at  J,  the  interval  be- 
tween two  successive  disappearances  of  the  satellite  in  entering,  or 
between  two  successive  reappearances  on  emerging  from  the  shadow 
is  found  to  be  about  42  hr.  30  min.  The  periodic  time  of  Jupiter 
being  about  11  yr.  10  mo.,  he  advances  but  a  short  distance,  as  to 
J',  while  the  earth  moves  to  E'  near  conjunction. 

Their  distance  is  now  increased  by  very  nearly  that  of  the  diam- 
eter of  the  earth's  orbit,  and  the  times  of  apparent  immersion  of  the 
satellite  are  delayed  beyond  the  computed  times  by  about  16  min. 
26  sec.  Since  the  periodic  time  of  the  satellite  is  constant,  Rosmer 


158  ELEMENTS    OF    WAVE    MOTION. 

therefore  concluded  that  light  required  16  min.  26  sec.  to  traverse- 
this  diameter. 

If  this  diameter  were  accurately  known,  and  the  exact  instant 
of  the  eclipse  could  be  noted,  a  very  nearly  exact  measure  for  tho 
velocity  of  light  could  be  computed.  The  reduction  of  more  than 
a  thousand  eclipses  of  Jupiter's  satellites,  by  Delambre,  gave  473.2 
mean  solar  seconds  for  the  time  of  travel,  which  corresponds  to  a 
solar  parallax  of  8.878",  and  to  a  velocity  of  298,793  kilometres  per 
second. 

258.  By  the  Aberration  of  Light.    Bradley,  in  1728, 
accounted  for  the  aberration  of  the  fixed  stars  by  assuming  that  the 
velocity  of  the  earth's  orbital  motion  had  an  appreciable  ratio  to  the 
velocity  of  light.     By  assuming  an  ideal  star  at  the  pole  of  the 
ecliptic,  the  value  of  the  constant  of  aberration,  according  to  his 
determination,  is  20.25",  which  corresponds  to  a  solar  parallax  of 
8.881".     According  to  W.  Struve,  this  constant  should  be  20.445", 
decreasing  the  parallax  to  8.797",  and  corresponding  to  a  velocity 
of  296,067  kilometres  per  second. 

The  principle  on  which  this  method  is  based  is  given  in  the  text 
on  Astronomy. 

259,  By  Actual  Measurement.     Owing  to  the  great 
velocity  of  light,  it  is  not  possible  to  measure  directly  the  very  small 
interval  of  time  required  for  light  to  traverse  any  terrestrial  dis- 
tance.     But  Fizeau,    Foucault,    Wheatstone,   Cornu,    and    more 
recently  Michelson,  have  succeeded  in  obtaining  its  value  within 
very  near  limits.     The  essential  principle  of  the  experiment  by 
Fizeau  consists  in  causing  a  toothed  wheel  to  revolve  with  great, 
but  uniform  velocity,  in  a  plane  perpendicular  to  the  track  of  a 
small  parallel  beam  of  light.     The  toothed  wheel  in  its  rotation 
alternately  permits  and  obstructs  the  passage  of  the  beam,  accord- 
ing as  an  interval  or  a  tooth  is  interposed  in  its  track.     The  beam 
of  light,  after  traversing  the  distance  determined  upon,  is  reflected 
by  a  small  mirror,  and  may  or  may  not  be  intercepted  on  its  return,, 
depending  on  the  ratio  of  the  velocity  of  rotation  of  the  wheel  and 
the  velocity  of  light.     Should  the  velocity  of  rotation  be  such  that 
the  returning  beam  passes  through  the  next  interval,  the  circum- 
ference of  the  wheel  would  have  moved  through  an  angle  equal  to 


RELATING    TO    SOUND    AND    LIGHT. 


159 


that  subtended  by  a  tooth  and  an  interval,  while  the  light  has  tra- 
versed double  the  distance  from  the  wheel  to  the  reflector. 

When  the  angular  velocity  of  the  wheel  is  doubled,  the  light 
passes  through  the  second  interval,  and  so  on.  The  value  for  the 
velocity  of  light  determined  by  this  method  is  315,364  kilometres. 
Cornu  has  recently  made  use  of  the  same  method,  but  with  a  very 
much  improved  apparatus,  and  has  found,  as  the  mean  of  504  ex- 
periments, the  value  of  300,400  kilometres  for  the  velocity  of  light 
in  vacuo,  with  a  probable  error  of  less  than  .001. 


Figure  a? 


260.  Foucault's  method  is  a  modification  of  the  preceding.  Let 
A,  in  Figure  37,  be  a  luminous  line,  BC  a  lens  whose  focal  length 
for  the  position  A  is  B#  +  bD,  bOc  a  revolving  mirror,  D  and  E 
circular  mirrors  whose  centre  is  at  0,  MM'  a  glass  plate,  R  a  reticle, 
and  L  an  eye  lens  to  view  the  image  of  A.  Now  if  the  mirror  0  is 
at  rest,  the  path  of  a  ray  from  A,  passing  through  the  lens  BC  and 
reflected  from  0,  is  AB#D ;  returning  by  reflection  from  D,  its  path 
is  DcCA.  A  part  of  its  light  is  reflected  from  the  first  surface  of 
MM',  and  the  image  of  A  is  seen  coincident  with  its  object  at  a. 
If  now  the  mirror  0  is  put  into  sufficiently  rapid  rotation,  the  re- 
turning ray  meets  it  at  b'Oc',  and  the  ray  is  reflected  along  cC'A', 
and  its  image  is  seen  at  «'.  The  angle  bOb'  is  known  from  the 
velocity  of  rotation,  the  distance  OD  is  given,  and  the  displacement 
aa  is  measured  by  a  micrometer. 

These  data  serve  to  measure  the  velocity  of  light  in  terms  of  the 


160  ELEMENTS    OF    WAVE    MOTION. 

angular  velocity  of  0.  By  the  addition  of  a  tube  filled  with  water 
at  FF',  the  velocity  of  light  in  water  was  found  and  shown  to  be 
less  than  that  in  air. 

In  the  diagram  annexed  to  the  figure,  ab  is  the  position  of  the 
image  when  0  is  at  rest,  c'  when  0  has  a  determinate  velocity,  and 
a'b'  the  corresponding  position  of  the  image  after  the  ray  has  tra- 
versed the  water.  The  result  of  this  determination  is  298,187 
kilometres  for  the  velocity  of  light,  corresponding  to  a  solar  parallax 
of  8.86".  Michelson,  by  an  ingenious  modification  of  the  method 
of  Foucault,  by  which  he  separated  his  mirrors  2000  feet,  and 
caused  one  of  them  to  revolve  257  times  per  second,  obtained  a  de- 
flection of  his  image  exceeding  133  millimetres,  and  thus  obtained 
results  which  are  claimed  to  be  exact  to  within  one  ten-thousandth, 
due  to  this  element  of  deflection. 

As  the  mean  of  1000  observations,  he  has  determined  299,930 
kilometres  per  second  for  the  velocity  of  light  in  vacuo. 

A  new  investigation  of  this  important  constant,  under  the  di- 
rection of  Prof.  Newcomb,  is  now  in  progress,  and  which,  when 
completed,  will  undoubtedly  be  as  close  an  approximation  to  the 
true  value  as  the  present  state  of  experimental  science  can  furnish. 

261.  Assuming  that  light  is  due  to  the  transversal  vibrations  of 
the  luminiferous  ether,  we  see,  Eq.  (119),  that  in  isotropic  media 
the  velocity  of  light  depends  on  the  coefficients  a,  b,  c,  etc.,  which 
are  functions  of  the  elasticity  and  density  of  the  medium. 

In  homogeneous  light,  or  that  in  which  A  is  constant,  V  will 
therefore  vary  when  light  passes  from  one  medium  into  another. 
The  conclusions  derived  by  supposing  a  variation  in  A,  the  medium 
remaining  the  same,  will  be  considered  under  the  dispersion  of  light. 


GEOMETRICAL    OPTICS. 

262,  In  geometrical  optics  it  is  only  necessary  to  take  account 
of  the  variation  of  the  velocity  due  to  a  change  in  the  elasticity  and 
density  of  the  ether,  in  passing  from  one  isotropic  medium  into 
another.  Hence,  we  consider  homogeneous  light  alone  in  the  dis- 
cussions which  follow.  These  changes  are  given  by  the  formula 

Y 

sin  0  =  fi  sin  0'  =  -=  sin  0'.  (289) 


RELATING    TO    SOUND    AND    LIGHT. 


161 


The  ratio  \i  is  called  the  index  of  refraction;  it  is  the  ratio  of 
the  velocity  of  light  propagation  in  the  two  media,  and  is  called  the 
absolute  index  when  the  medium  from  which  it  passes  is  the  ether. 
When  any  two  other  velocities  are  compared,  the  ratio  is  called  the 
relative  index ;  the  relative  index  is  then  only  the  ratio  of  the  two 
absolute  indices.  When  reflection  is  considered  as  a  particular  case 
of  refraction,  \i  is  always  taken  as  —  1. 

263,  A  radiant  is  a  point  from  which  the  rays  proceed ;  it  is 
said  to  be  real  when  the  beam  is  parallel  or  diverging,  and  virtual 
when  converging.     A  focus  is  the  point  in  which  the  rays  meet 
after  deviation,  or  in  which  they  would  meet  if  prolonged  in  either 
direction  ;  in  the  former  case  the  focus  is  real,  and  in  the  latter,  if 
the  point  of  meeting  is  found  by  prolonging  the  rays  backward,  it  is 
virtual.     A  radiant  and  its  focus  are  the  centres  of  curvature  of  the 
nndeviated  and  deviated  pencils,  respectively.     In   the   following 
discussions,  distances  estimated  in  the  direction  of  wave  propaga- 
tion, from  any  origin  whatever,  are  taken  as  negative,  and  iu  the 
contrary  direction  as  positive. 

264.  Deviation  of  Light  by  Plane  Surfaces.    Let  us 

suppose  the  incident  medium  to  be  any  whatever,  as  air,  and  that 
the  ray  enters  any  other  medium,  as  glass,  whose  surface  is  plane, 
Then,  Figure  38,  we  have,  for  the  first  refraction, 


sin  0  =  \i  sin  0', 


(290) 


in  which  \i  is  the  relative  index  of  air  referred  to  glass ;  and  for  the 
first  reflection  we  have 

sin  0  =  —  sin  0.  (291) 

The  angle  0'  is  less  than  0,  because 
^  =  -y  is  greater  than  unity,  since  the 

velocity  of  wave  propagation  of  light  in 
air  is  found  by  experiment  to  be  greater 
than  that  in  glass.  Should  the  velocity 
in  the  medium  of  intromittance  be 
greater  than  that  in  the  medium  of  in- 
cidence, \i  would  be  less  than  unity  and 
<f>'  would  be  greater  than  0.  The  re-  Figure  38. 


